Quantifying Bimodality Part 2: A Likelihood Ratio Test for the Comparison of a Unimodal Normal Distribution and a Bimodal Mixture of Two Normal Distributions. Bruno D. Zumbo is

Scientists in a variety of fields are often faced with the question of whether a sample is best described as unimodal or bimodal. In an earlier paper (Frankland & Zumbo, 2002), a simple and convenient method for assessing bimodality was described. That method is extended by developing and demonstrating a likelihood ratio test (LRT) for bimodality for the comparison of a unimodal normal distribution and a bimodal mixture of two normal distributions. As in Frankland and Zumbo (2002), the LRT approach is demonstrated using algorithms in SPSS

Quantifying Bimodality Part I: An Easily Implemented Method Using \u3cem\u3eSPSS\u3c/em\u3e

Scientists in a variety of fields are faced with the question of whether or not a particular sample of data are best described as unimodal or bimodal. We provide a simple and convenient method for assessing bimodality. The use of the non-linear algorithms in SPSS for modeling complex mixture distributions is demonstrated on a unimodal normal distribution (with 2 free parameters) and on bimodal mixture of two normal distributions (with 5 free parameters)

Yield scaling, size hierarchy and fluctuations of observables in fragmentation of excited heavy nuclei

Multifragmentation properties measured with INDRA are studied for single sources produced in Xe+Sn reactions in the incident energy range 32-50 A MeV and quasiprojectiles from Au+Au collisions at 80 A MeV. A comparison for both types of sources is presented concerning Fisher scaling, Zipf law, fragment size and fluctuation observables. A Fisher scaling is observed for all the data. The pseudo-critical energies extracted from the Fisher scaling are consistent between Xe+Sn central collisions and Au quasi-projectiles. In the latter case it also corresponds to the energy region at which fluctuations are maximal. The critical energies deduced from the Zipf analysis are higher than those from the Fisher analysis.Comment: 30 pages, accepted for publication in Nuclear Physics A, references correcte

Fragment size correlations in finite systems - application to nuclear multifragmentation

We present a new method for the calculation of fragment size correlations in a discrete finite system in which correlations explicitly due to the finite extent of the system are suppressed. To this end, we introduce a combinatorial model, which describes the fragmentation of a finite system as a sequence of independent random emissions of fragments. The sequence is accepted when the sum of the sizes is equal to the total size. The parameters of the model, which may be used to calculate all partition probabilities, are the intrinsic probabilities associated with the fragments. Any fragment size correlation function can be built by calculating the ratio between the partition probabilities in the data sample (resulting from an experiment or from a Monte Carlo simulation) and the 'independent emission' model partition probabilities. This technique is applied to charge correlations introduced by Moretto and collaborators. It is shown that the percolation and the nuclear statistical multifragmentaion model ({\sc smm}) are almost independent emission models whereas the nuclear spinodal decomposition model ({\sc bob}) shows strong correlations corresponding to the break-up of the hot dilute nucleus into nearly equal size fragments

Pion radii in nonlocal chiral quark model

The electromagnetic radius of the charged pion and the transition radius of the neutral pion are calculated in the framework of the nonlocal chiral quark model. It is shown in this model that the contributions of vector mesons to the pion radii are noticeably suppressed in comparison with a similar contribution in the local Nambu--Jona-Lasinio model. The form-factor for the process gamma*pi+pi- is calculated for the -1 GeV^2<q^2<1.6 GeV^2. Our results are in satisfactory agreement with experimental data.Comment: 7 pages, 7 figure

Transition from participant to spectator fragmentation in Au+Au reaction between 60 AMeV and 150 AMeV

Using the quantum molecular dynamics approach, we analyze the results of the recent INDRA Au+Au experiments at GSI in the energy range between 60 AMeV and 150 AMeV. It turns out that in this energy region the transition toward a participant-spectator scenario takes place. The large Au+Au system displays in the simulations as in the experiment simultaneously dynamical and statistical behavior which we analyze in detail: The composition of fragments close to midrapidity follows statistical laws and the system shows bi-modality, i.e. a sudden transition between different fragmentation pattern as a function of the centrality as expected for a phase transition. The fragment spectra at small and large rapidities, on the other hand, are determined by dynamics and the system as a whole does not come to equilibrium, an observation which is confirmed by FOPI experiments for the same system.Comment: published versio

Statistical Multifragmentation of Non-Spherical Expanding Sources in Central Heavy-Ion Collisions

We study the anisotropy effects measured with INDRA at GSI in central collisions of Xe+Sn at 50 A.MeV and Au+Au at 60, 80, 100 A.MeV incident energy. The microcanonical multifragmentation model with non-spherical sources is used to simulate an incomplete shape relaxation of the multifragmenting system. This model is employed to interpret observed anisotropic distributions in the fragment size and mean kinetic energy. The data can be well reproduced if an expanding prolate source aligned along the beam direction is assumed. An either non-Hubblean or non-isotropic radial expansion is required to describe the fragment kinetic energies and their anisotropy. The qualitative similarity of the results for the studied reactions suggests that the concept of a longitudinally elongated freeze-out configuration is generally applicable for central collisions of heavy systems. The deformation decreases slightly with increasing beam energy.Comment: 35 pages, 19 figures, submitted to Nuclear Physics

Multiplicity correlations of intermediate-mass fragments with pions and fast protons in 12C + 197Au

Low-energy pi+ (E < 35 MeV) from 12C+197Au collisions at incident energies from 300 to 1800 MeV per nucleon were detected with the Si-Si(Li)-CsI(Tl) calibration telescopes of the INDRA multidetector. The inclusive angular distributions are approximately isotropic, consistent with multiple rescattering in the target spectator. The multiplicity correlations of the low-energy pions and of energetic protons (E > 150 MeV) with intermediate-mass fragments were determined from the measured coincidence data. The deduced correlation functions 1 + R \approx 1.3 for inclusive event samples reflect the strong correlations evident from the common impact-parameter dependence of the considered multiplicities. For narrow impact-parameter bins (based on charged-particle multiplicity), the correlation functions are close to unity and do not indicate strong additional correlations. Only for pions at high particle multiplicities (central collisions) a weak anticorrelation is observed, probably due to a limited competition between these emissions. Overall, the results are consistent with the equilibrium assumption made in statistical multifragmentation scenarios. Predictions obtained with intranuclear cascade models coupled to the Statistical Multifragmentation Model are in good agreement with the experimental data.Comment: 9 pages, 11 figures, subm. to EPJ

Bimodality - a general feature of heavy ion reactions

Recently, is has been observed that events with the {\it same} total transverse energy of light charged particles (LCP) in the quasi target region, $E_{\perp 12}^{QT}$, show two quite distinct reaction scenarios in the projectile domain: multifragmentation and residue production. This phenomenon has been dubbed "bimodality". Using Quantum Molecular Dynamics calculations we demonstrate that this observation is very general. It appears in collisions of all symmetric systems larger than Ca and at beam energies between 50 A.MeV and 600 A.MeV and is due to large fluctuations of the impact parameter for a given $E_{\perp 12}^{QT}$. Investigating in detail the $E_{\perp 12}^{QT}$ bin in which both scenarios are present, we find that neither the average fragment momenta nor the average transverse and longitudinal energies of fragments show the behavior expected from a system in statistical equilibrium, in experiment as well as in QMD simulations. On the contrary, the experimental as well as the theoretical results point towards a fast process. This observation questions the conjecture that the observed bimodality is due to the coexistence of 2 phases at a given temperature in finite systems.Comment: accepted PR

Gross Properties and Isotopic Phenomena in Spectator Fragmentation

A systematic study of isotopic effects in the break-up of projectile spectators at relativistic energies has been performed with the ALADiN spectrometer at the GSI laboratory. Searching for signals of criticality in the fragment production we have applied the model independent universal fluctuations theory already proposed to track criticality signals in multifragmentation to our data. The fluctuation of the largest fragment charge and of the asymmetry of the two and three largest fragments and their bimodal distribution have also been analysed.Comment: 6 pages, 4 figures, IX International Conference on Nucleus-Nucleus Collisions, Rio de Janeiro, Brazil, August 28 - September 1, 200