Isoscaling Parameters Research Articles

Isocaling and the symmetry energy in the multifragmentation regime of heavy-ion collisions

The ratio of the symmetry energy coefficient to temperature, $a_sym/T$, in Fermi energy heavy ion collisions, has been experimentally extracted as a function of the fragment atomic number using isoscaling parameters and the variance of the isotope distributions. The extracted values have been compared to the results of calculations made with an Antisymmetrized Molecular Dynamics (AMD) model employing a statistical decay code to account for deexcitation of excited primary fragments. The experimental values are in good agreement with the values calculated but are significantly different from those characterizing the yields of the primary AMD fragments.

Isoscaling of mass [formula omitted] reconstructed quasiprojectiles from collisions in the Fermi energy regime

Isoscaling studies of fragments with Z = 1 – 8 from reconstructed quasiprojectiles of mass A ≃ 40 from the Ar 40 + Sn 124 , 112Sn and Ca 40 + Sn 124 , 112Sn reactions at beam energy of 45 MeV/nucleon were performed. After initial efforts to obtain isoscaling with the “traditional” approach of using pairs of systems differing in their isospin asymmetry (or neutron-to-proton ratio N / Z ), “intra-system” isoscaling for each of these four systems was obtained using fragment sources restricted in two narrow N / Z regions (one neutron-rich and one neutron-poor). The observed isoscaling behavior was excellent and the isoscaling parameter α was found to decrease with increasing excitation energy. Corrections due to undetected neutrons were also taken into account in the source N / Z determination by using the theoretical models DIT (Deep Inelastic Transfer) and SMM (Statistical Multifragmentation Model) along with a software replica of the experimental setup. These corrections were applied to the determination of the parameter Δ (expressing the difference in the isospin asymmetry of the two sources used in the isoscaling). The reduced isoscaling parameter α / Δ was obtained and found to decrease as the excitation energy of the quasiprojectile source increases, in good agreement with recent work on reconstructed mass A ≃ 80 quasiprojectiles. This decrease of α / Δ may point to a decrease of the symmetry energy coefficient with increasing excitation energy.

Symmetry energy and the isoscaling in reactions on enriched tin isotopes

The coefficients of symmetry energy term for fragments with Z=4,11,12 measured in multifragmentation reactions initiated by proton and deuteron with energy of 3.65A GeV on enriched tin isotopes 112,118,120,124Sn are determined. The dependence of isoscaling parameter on the excitation energy, the temperature of fragmenting systems and the density ratio for heavy mass products are analised.

Isospin effects and the density dependence of the nuclear symmetry energy

The density dependence of the nuclear symmetry energy is inspected using the statistical multifragmentation model with Skyrme effective interactions. The model consistently considers the expansion of the fragments' volumes at finite temperature at the freeze-out stage. By selecting parametrizations of the Skyrme force that lead to very different equations of state for the symmetry energy, we investigate the sensitivity of the isoscaling parameter and the isotopic distributions to differences in the symmetry energy. Our results suggest that, in spite of being sensitive to the thermal dilation of the fragments' volumes, it is difficult to distinguish among the Skyrme forces from the isoscaling analysis. On the other hand, the isotopic distribution of the emitted fragments turns out to be very sensitive to the force employed in the calculation.

Surface and symmetry energies in isoscaling for multifragmentation reactions

The possibilities of modifications for symmetry and surface energy coefficients of nuclear matter at freeze-out density are investigated theoretically by means of isoscaling (isotopic scaling) on the basis of a statistical multifragmentation model. It is seen that while the isoscaling parameters as predicted by Markov chain calculations for the single sources 124Sn and 112Sn are very sensitive to the symmetry term, the surface energy variation slightly affects the isoscaling parameters in the multifragmentation region. The influence of these changes on the mean neutron to proton ratios of light fragments for 2 ⩽ Z ⩽ 10 is also shown. Comparing our results with MSU experimental data, it is confirmed that a significant reduction of the symmetry-term coefficient is found necessary to reproduce the isoscaling parameters of Z ⩽ 8 fragments.

Temperature effects in nuclear isoscaling

The properties of the nuclear isoscaling at finite temperature are investigated and the extent to which its parameter $\ensuremath{\alpha}$ holds information on the symmetry energy is examined. We show that, although finite temperature effects invalidate the analytical formulas that relate the isoscaling parameter $\ensuremath{\alpha}$ to those of the mass formula, the symmetry energy remains the main ingredient that dictates the behavior of $\ensuremath{\alpha}$ at finite temperatures, even for very different sources. This conclusion is not obvious as it is not true in the vanishing temperature limit, where analytical formulas are available. Our results also reveal that different statistical ensembles lead to essentially the same conclusions based on the isoscaling analysis, for the temperatures usually assumed in theoretical calculations in the nuclear multifragmentation process.

Isoscaling Behavior in 48,40Ca+9Be Collisions at Intermediate Energy Investigated by the HIPSE Model

The fragment production cross sections for 140 MeV/nucleon 48,40Ca+9Be reactions are calculated by the heavyion phase space exploration (HIPSE) model. Isoscaling behavior is observed. The isoscaling parameters α and β for both heavy and light fragments from the HIPSE model calculations are in good agreement with the experimental data. It is shown that the potential parameters in the HIPSE model have very little effect on the isoscaling parameters. The effect of the excitation energy and evaporation on the isoscaling behavior is also discussed.

Isoscaling of fragments withZ=1−17from reconstructed quasiprojectiles

In heavy-ion collisions, isoscaling provides a method for studying the evolution of nuclear symmetry energy as a function of excitation energy. One challenge in using isoscaling is to accurately determine the neutron-to-proton ratio ($N/Z$) of the fragmenting source. Isoscaling results are presented for the reactions of $^{86,78}\mathrm{Kr}+^{64,58}\mathrm{Ni}$ at 35 MeV/nucleon taken on the NIMROD-ISiS array at Texas A University. The $N/Z$ of the source was calculated from the isotopically identified fragments and experimentally measured neutrons emitted from reconstructed quasiprojectiles. These data exhibit isoscaling for elements with $Z=1\text{\ensuremath{-}}17$ over a broad range of isotopes. The isoscaling parameter $\ensuremath{\alpha}$ is shown to increase with increasing difference in the neutron composition (\ensuremath{\Delta}) of the compared sources. For a selected \ensuremath{\Delta}, the ratio $\ensuremath{\alpha}$/\ensuremath{\Delta} is also shown to decrease with increasing excitation energy. This may reflect a corresponding decrease in the nuclear symmetry energy.

Isoscaling in statistical fragment emission in an extended compound nucleus model

Based on an extended compound nucleus model, isospin effects in statistical fragment emission from excited nuclear systems are investigated. An experimentally observed scaling behavior of the ratio of isotope yields Yi(N,Z) from two similar emitting sources with different neutron-to-proton ratios is predicted theoretically, i.e., the relationship of Y2/Y1∝exp(αN+βZ) is demonstrated. The symmetry energy coefficient Csym extracted from the simulation results is ∼27 MeV which is consistent with realistic theoretical estimates and recent experimental data. The influence of the surface entropy on the isoscaling behavior is discussed in detail. It is found that although the surface entropy increases the numerical values of isoscaling parameters α and β, it does not affect the isoscaling behavior qualitatively and has only a minor effect on the extracted symmetry energy coefficient.

Isoscaling behavior studied by HIPSE model

The isoscaling behavior in the reaction system of 58,64Ni + 9Be has been studied by using the heavy-ion phase-space exploration(HIPSE) model. The extracted isoscaling parameters α and β for both heavy and light fragments for HIPSE model calculations are in good agreement with recent experimental data. The investigation shows that the parameters in the HIPSE model have some effect on the isoscaling parameter. The isoscaling parameters for hot and cold fragments have been extracted.

ISOSPIN EFFECT AND ISOSCALING PHENOMENON IN PROJECTILE FRAGMENTATION

We have studied the isospin effect and isoscaling behavior in projectile fragmentation using a modified statistical abrasion-ablation (SAA) model. The relationship between neutron skin thickness (δnp) and neutron abrasion cross section (σ nabr ) has also been investigated. We find the normalized peak differences and reduced isoscaling parameters decrease with (Z proj - Z)/Z proj or the excitation energy per nucleon and have no significant dependence on the size of the reaction systems. The excitation energy dependence of the symmetry energy coefficients are tentatively extracted from α and β. Assuming Fermi-type density distributions for proton and neutron and introducing a parameter to adjust the diffuseness of the neutron density distribution for neutron-rich nucleus, δnp and σ nabr for neutron-rich nucleus were found to have a linear correlation. It is suggested that σ nabr could be used as an observable to extract the neutron skin thickness for neutron-rich nucleus.

Isoscaling and the high-temperature limit

This study shows that isoscaling, usually studied in nuclear reactions, is a phenomenon common to all cases of fair sampling. Exact expressions for the yield ratio R{sub 21} and approximate expressions for the isoscaling parameters {alpha} and {beta} are obtained and compared to experimental results. It is concluded that nuclear isoscaling is bound to contain a component due to sampling and, thus, a word of caution is issued to those interested in extracting information about the nuclear equation of state from isoscaling.

Isoscaling, symmetry energy and thermodynamic models

The isoscaling parameter usually denoted by α depends upon both the symmetry energy coefficient and the isotopic contents of the dissociating systems. We compute α in theoretical models: first in a simple mean field model and then in thermodynamic models using both grand canonical and canonical ensembles. For finite systems the canonical ensemble is much more appropriate. The model values of α are compared with a much used standard formula. Next we turn to cases where in experiments, there are significant deviations from isoscaling. We show that in such cases, although the grand canonical model fails, the canonical model is capable of explaining the data.

Influence of in-medium nucleon–nucleon cross section on the isoscaling parameter α

The influence of in-medium nucleon–nucleon cross section on the isoscaling parameter α is investigated for two couples of central nuclear reactions Ca 40 + Ca 40 and Ca 60 + Ca 40 ; Sn 112 + Sn 112 and Sn 124 + Sn 124 within the isospin dependent quantum molecular dynamics. The calculated result shows that the influence of the in-medium nucleon–nucleon cross section on the isoscaling parameter α is mainly determined by the corresponding number of collisions, both for isospin dependent and isospin independent parameterizations. The mechanisms behind the effects of the in-medium nucleon–nucleon cross sections on the α are investigated in more details.

Probing the symmetry energy from the nuclear isoscaling

Using different parametrizations of the nuclear mass formula, we study the sensitivity of the isoscaling parameters to the mass formula employed in grand-canonical calculations. Previous works on isoscaling have suggested that the symmetry energy implied in such calculations is anomalously smaller than that suggested by fits to nuclear masses. We show that surface corrections to the symmetry energy naturally broadens the isotopic distribution thus allowing for values of the symmetry energy which more closely match those obtained from nuclear masses.

MEDIUM EFFECT OF NUCLEON–NUCLEON CROSS SECTION ON THE ISOSCALING PARAMETER α

The medium effect of in-medium nucleon–nucleon cross section [Formula: see text] on the isoscaling parameter α is investigated for two couples of central nuclear reactions 40 Ca +48 Ca and 60 Ca +48 Ca ; 112 Sn +112 Sn and 124 Sn +124 Sn at beam energy region from 40 to 60 MeV/nucleon with isospin dependent quantum molecular dynamics. It is found that there is the obvious medium effect of [Formula: see text] on the isoscaling parameters α. The mechanism for the medium effect of [Formula: see text] on α is investigated.

Density and Symmetric Potential Dependences of Isoscaling Behaviour in the Lattice Gas Model

Isoscaling behaviour of the statistical emission fragments from the equilibrated sources with Z = 30 and N = 30, 33, 36 and 39 is investigated in the framework of the isospin-dependent lattice gas model. The dependences of isoscaling parameters a on source isospin asymmetry, temperature and freeze-out density are studied, and the ‘symmetry energy’ is deduced from isoscaling parameters. The results show that symmetry energy Csym is insensitive to the change of temperature but follows the power-law dependence on the freeze-out density ρ. The effect of strength of asymmetry of nucleon-nucleon interaction potential on the density dependence of the symmetry energy is discussed.

INFLUENCES OF THE ISOSPIN-DEPENDENCE OF IN-MEDIUM NUCLEON–NUCLEON CROSS-SECTION AND MOMENTUM DEPENDENT INTERACTION ON ISOSCALING PARAMETER

Influences of the isospin-dependent in-medium nucleon–nucleon cross-section [Formula: see text] and momentum-dependent interaction (MDI) on the isoscaling parameter α are investigated for two central collisions 40 Ca + 40 Ca and 60 Ca + 60 Ca . These collisions are with isospin dependent quantum molecular dynamics in the beam energy region from 40 to 60 MeV/nucleon. The isotope yield ratio R21(N, Z) for the above two central collisions depends exponentially on the neutron number N and proton number Z of isotopes, with an isoscaling. In particular, the isospin-dependent [Formula: see text] and MDI induce an obvious decrease of the isoscaling parameter α. The mechanism of the decreases of α by both [Formula: see text] and MDI are studied respectively.

Medium effect of nucleon-nucleon cross section on the isoscaling parameter α

The medium effect of nucleon-nucleon cross section σmedNN(α) on the isoscaling parameter α is investigated for two couples of central nuclear center reactions 40Ca+48Ca and 60Ca+48Ca， 112Sn+112Sn and 124Sn+124Sn with isospin dependent quantum molecular dynamics at beam nucleon energies from 40 to 60MeV. It is found that there is very obvious medium effects of nucleon-nucleon cross section σmedNN(α) on the isoscaling parameters α. The mechanism of the above properties is also studied and discussed.

Isospin effect of nucleon-nucleon cross section on the isoscaling parameter α

The isospin effects of isoscaling parameter α on the nucleon-nucleon cross section are investigated for two couples of central nuclear reactions 40C+ 40Ca with 60Ca + 40Ca and 112Sn+ 112Sn with 124Sn+ 124Sn within an isospin dependent quantum molecular dynamics. It is found that the isospin effect of isoscaling parameter α on the isospin dependent nucleon-nucleon cross section σmedNN(αm) compared with that on the isospin independent nucleon-nucleon cross sections σ1NN(αm) is very obvious. However, the isospin effect of isoscaling parameter α on σmedNN(αm) compared with that on the isospin independent nucleon-nucleon cross section σ2NN(αm) is small. The mechanisms for the above difference are investigated in detail.