Isospin diffusion from Ca 40 , 48 + Ca 40 , 48
Q. FableL. BaldesiS. BarliniE. BonnetB. BorderieR. BougaultA. CamaianiG. CasiniA. ChbihiC. CiampiJ. A. DueñasJ. D. FranklandT. GénardD. GruyerM. HenriB. HongS. KimA.J. KordyaszT. KozikA. Le FèvreN. Le NeindreI. LombardoO. LopezT. MarchiP. MariniS. H. NamAkira OnoJun‐Hyoung ParkM. PârlogS. PiantelliA. Rebillard-SouliéG. VerdeE. Vient
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This article presents an investigation of isospin equilibration in cross-bombarding $^{40,48}\mathrm{Ca}+^{40,48}\mathrm{Ca}$ reactions at 35 MeV/nucleon, by comparing experimental data with filtered transport model calculations. Isospin diffusion is studied using the evolution of the isospin transport ratio with centrality. The asymmetry parameter $\ensuremath{\delta}=(N\ensuremath{-}Z)/A$ of the quasiprojectile (QP) residue is used as isospin-sensitive observable, while a recent method for impact parameter reconstruction is used for centrality sorting. A benchmark of global observables is proposed to assess the relevance of the antisymmetrized molecular dynamics (amd) model, coupled to gemini$++$, in the study of dissipative collisions. Our results demonstrate the importance of considering cluster formation to reproduce observables used for isospin transport and centrality studies. Within the amd model, we prove the applicability of the impact parameter reconstruction method, enabling a direct comparison to the experimental data for the investigation of isospin diffusion. For both, we evidence a tendency to isospin equilibration with an impact parameter decreasing from 9 to 3 fm, while the full equilibration is not reached. A weak sensitivity to the stiffness of the equation of state employed in the model is also observed, with a better reproduction of the experimental trend for the neutron-rich reactions.Fragment yield data, selected with different values of the isospin asymmetry of the quasiprojectile from 78, 86 Kr + 58, 64 Ni reactions at E lab = 35 MeV/nucleon, have been analyzed within the framework of Landau free energy approach. Fits to the free energy data indicated the system to be in the regime of a first-order phase transition. The plot of free energy versus isospin asymmetry of the fragments showed a systematic change with increasing isospin asymmetry of the fragmenting source. This observation demonstrated the role of quasiprojectile isospin asymmetry as an external field which governed the minima positions in the free energy plot. The position of the central minimum was found to be in close agreement with the average isospin asymmetry of the fragments calculated excluding neutrons and protons. Significant deviations from the free energy plot were observed for N = Z fragments, particularly for the lighter ones, which could be corrected after inclusion of odd–even effects in the analysis. Analysis of yield data of N = Z nuclei gave a reasonable estimate of temperature. Analysis of free energy data gated with excitation energy of the fragmenting source showed a dependence of fit parameters of Landau free energy equation on the excitation energy as well as isospin asymmetry of the source.
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We investigate the role of isospin momentum-dependent interaction on the isospin fractionation ratio and its dynamical mechanism in the intermediate energy heavy ion collisions, by inserting an isospin degree of freedom into the momentum-dependent interaction to obtain an isospin momentum-dependent interaction given in a form practically usable in the isospin-dependent quantum molecular dynamics model. It is found that the isospin momentum-dependent interaction brings an important isospin effect into the isospin fractionation ratio. In particular, the isospin momentum-dependent interaction reduces obviously the reduction of isospin fractionation ratio. Thus the isospin dependence of momentum-dependent interaction is thus important for studying accurately the equation of state of isospin asymmetry nuclear matter.
Momentum (technical analysis)
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The relation between strong isospin I and weak isospin i is discussed. In particular an equation between the third components I 3 and i 3 is given. This relation indicates that the strong isospin and weak isospin symmetries are both SU (2) subgroups of a new SU (3) symmetry underlying the structure of leptons and quarks.
Weak isospin
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We obtain two isospin amplitude decompositions of the B→Kππ decays. We use the method of addition of three isospin vectors, and prove the equivalency of the two isospin amplitude decompositions obtained. We only have considered contributions of the Hamiltonian to the transitions with change of isospin ΔI=0 and ΔI=1. The symmetry of isospin allows to relate the charged B+→K+π+π−, B+→K+π0π0, and B+→K0π0π+ channels with the neutral B0→K0π+π−, B0→K0π0π0, and B0→K+π0π− channels. Additionally, we obtain equivalent triangular relations in the charged and neutral channels.
Hamiltonian (control theory)
Weak isospin
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Abstract In statistical hydrology, three different methods have been used to calculate the expected values of observable quantities in an ensemble. The first is to use an observable as a parameter to characterize the “states” of the ensemble and then to calculate the average value of the observable over all possible states; this is the expected value. The second method is to use an observable as a parameter to characterize the “states” and then to designate the most probable value of the observable as the expected value; this is known as the “minimum variance principle”. Finally, the third method is to use a parameter different from an observable to characterize the “states” and then to take the value of the observable for the most probable state as the expected value. Of all these different methods, only the first one is, in principle, physically correct. Thus, a study was made to determine the situations where methods other than the first can be used to calculate the expectation values of observables. It was found that this can be done only in those situations where all of the following conditions hold: (i) The number of parameters used to characterize the states is finite; (ii) these parameters are either the same as the observables or are explicitly related to the observables; and (iii) each of the parameters has a symmetrical and unimodal probability distribution. The cases where the minimum variance principle is valid, are thereby delineated.
Value (mathematics)
Expected value
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Fluctuations in small biological systems can be crucial for their function. Large-deviation theory characterizes such rare events from the perspective of stochastic processes. In most cases it is very difficult to directly determine the large-deviation functions. Circumventing this necessity, I present a method to quantify the fluctuation spectra for arbitrary Markovian models with finite state space. Under non-equilibrium conditions, current-like observables are of special interest. The space of all current-like observables has a natural decomposition into orthogonal complements. Remarkably, the fluctuation spectrum of any observable is entirely determined by only one of these components. The method is applied to study differences of fluctuations in setups sampling the same dynamics at different resolutions. Coarse graining relates these models and can be done in a way that preserves expectation values of observables. However, the effects of the coarse graining on the fluctuations are not obvious. These differences are explicitly worked out for a simple model system.
Granularity
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We study the isospin splitting of the giant resonance of $T=\frac{1}{2}$ nuclei with sum rules. The particular isospin geometry involved in this case permits a number of simplifications in the isospin sum rules and several largely model independent relations are obtained and discussed. It turns out that for several light isodoublets the "symmetry" energy $U$ tends to be smaller that $60(T+1){A}^{\ensuremath{-}1}$ MeV.NUCLEAR STRUCTURE $T=\frac{1}{2}$ nuclei; calculated isospin splitting of $E1$ resonance with sum rules, model independent results discussed.
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