A generalized thermodynamics for power-law statistics

We show that there exists a natural way to define a condition of generalized thermal equilibrium between systems governed by Tsallis thermostatistics, under the hypotheses that (i) the coupling between the systems is weak, (ii) the structure functions of the systems have a power-law dependence on the energy. It is found that the q values of two such systems at equilibrium must satisfy a relationship involving the respective numbers of degrees of freedom. The physical properties of a Tsallis distribution can be conveniently characterized by a new parameter η which can vary between 0 and +∞, these limits corresponding, respectively, to the two opposite situations of a microcanonical distribution and of a distribution with a predominant power-tail at high energies. We prove that the statistical expression of the thermodynamic functions is univocally determined by the requirements that (a) systems at thermal equilibrium have the same temperature, (b) the definitions of temperature and entropy are consistent with the second law of thermodynamics. We find that, for systems satisfying the hypotheses (i) and (ii) specified above, the thermodynamic entropy is given by Rényi entropy.     

The statistical theory of multi-step compound and direct reactions

The theory of nuclear reactions is extended so as to include a statistical treatment of multi-step processes. Two types are distinguished, the multi-step compound and the multi-step direct. The wave functions for the system are grouped according to their complexity. The multi-step direct process involves explicitly those states which are open, while the multi-step compound involves those which are bound. In addition to the random phase assumption which is applied differently to the multi-step direct and to the multi-step compound cross-sections, it is assumed that the residual interaction will have non-vanishing matrix elements between states whose complexities differ by at most one unit. This is referred to as the chaining hypothesis. Explicit expressions for the double differential cross-section giving the angular distribution and energy spectrum are obtained for both reaction types. The statistical multi-step compound cross-sections are symmetric about 90°. The classical statistical theory of nuclear reactions is a special limiting case. The cross-section for the statistical multi-step direct reaction consists of a set of convolutions of single-step direct cross-sections. For the many step case it is possible to derive a diffusion equation in momentum space. Application is made to the reaction ¹⁸¹Ta(p, n)¹⁸¹W using the statistical multi-step compound formalism.     

Ericson fluctuations and the distribution of randomS-matrix elements

Starting from a random matrix model for nuclear reactions, we show that the fluctuatingS-matrix elements in general have a non-gaussian distribution. The consequences of this for Ericson fluctuations are pointed out. In the absence of direct reactions symmetry properties of the distribution function lead to simple expressions for the cross section correlation functions. In particular a formula for calculating the cross section auto-correlation function is presented.     

Incomplete fusion reactions: Analysis of excitation functions and recoil range distributions in 16O + 51V

Incomplete fusion reactions were investigated by measuring the excitation functions of nine evaporation residues in ¹⁶O + ⁵¹V reaction in the beam energy 4-6 MeV/amu, using the well-known recoil catcher technique and gamma-ray spectrometry. The experimental data were compared with that obtained from Monte Carlo simulation calculations using the PACE2 code. The results indicate the presence of incomplete fusion process in the production of two alpha emission products. This was further confirmed by the measurement of recoil range distribution of these isotopes at 96 MeV beam energy. Calculations of the average angular momentum associated with these products revealed the peripheral nature of these ICF reactions. Received: 20 June 2001 / Accepted: 11 September 2001     

Stability of incomplete entropy and incomplete expectation

Based on the principle of maximum entropy, the q-exponential distribution can be derived from several different nonextensive entropies including the incomplete entropy. It is widely used in nonextensive statistical mechanics. In the present paper, it is shown that the incomplete expectation value and incomplete entropy are stable under small deformation of the probability distribution function of the system.     

Constraining the nuclear gluon distribution in eA processes at RHIC

A systematic determination of the gluon distribution is of fundamental interest in understanding the parton structure of nuclei and the QCD dynamics. Currently, the behavior of this distribution at small x   (high energy) is completely undefined. In this Letter we analyze the possibility of constraining the nuclear effects present in xgA$x{g}_A$ using the inclusive observables which would be measured in the future electron-nucleus collider at RHIC. We demonstrate that the study of nuclear longitudinal and charm structure functions allows to estimate the magnitude of shadowing and antishadowing effects in the nuclear gluon distribution.     

Cross sections for nuclear reactions in collisions of238U+238U and238U +197Au near and below the coulomb barrier

Cross sections for nuclear reactions at beam energies near and below the spherical Coulomb barrier V _c were measured in the very heavy collision systems²³⁸U +²³⁸U and²³⁸U +¹⁹⁷Au. The most probable reaction channel with mass transfer is the one-neutron transfer. Its excitation function is understood in terms of Rutherford trajectories together with the quantal process of neutron tunnelling over large distances. In addition, the exchange of up to 15 nucleons is observed down to 0.90 V _c . The excitation functions for the multi-nucleon transfer products have much steeper slopes than that for one-neutron transfer, and are steeper for²³⁸U +¹⁹⁷Au than for²³⁸U +²³⁸U, suggesting that nuclear contact is established in the associated collisions. The angular distribution for one selected multi-nucleon transfer product,²²⁷Th, shows that its formation occurs in more central collisions within contact times shorter than about 10^?21 s. There is no evidence for very longlived di-nuclear systems in the these reactions.     

Energy dependence of Lévy stability and multi-fractal spectrum in e<Superscript>+</Superscript>e<Superscript>−</Superscript> collisions

Various nuclear reactions like quasi-fission, fusion-fission or particle and cluster evaporation from excited compound nuclei were studied in heavy-ion reactions at the velocity filter SHIP of GSI. The velocity filter offers the possibility to detect all reaction products under zero degree relative to the beam direction. Together with the measurement of the product velocity distribution this allows for an identification of the underlying reaction mechanism. This article is focussed on reactions of ²⁵Mg and ⁶⁴Ni beams on ^{206, 207}Pb targets at energies of 5.9×AMeV and 8.7×AMeV . Besides evaporation residues from ²⁵Mg + ²⁰⁶Pb collisions we found evidence for rotation and quasi-fission of nuclear molecules formed in the entrance channel after the capture stage. The break-up of the systems showed a preferred clustering leading to isotopes in the region 84 ⩽ Z ⩽ 88 and 122 ⩽ N ⩽ 127 of the chart of nuclei.     

On the construction of complex networks with optimal Tsallis entropy

In this work, we first formulate the Tsallis entropy in the context of complex networks. We then propose a network construction whose topology maximizes the Tsallis entropy. The growing network model has two main ingredients: copy process and random attachment mechanism (C-R model). We show that the resulting degree distribution exactly agrees with the required degree distribution that maximizes the Tsallis entropy. We also provide another example of network model using a combination of preferential and random attachment mechanisms (P-R model) and compare it with the distribution of the Tsallis entropy. In this case, we show that by adequately identifying the exponent factor q, the degree distribution can also be written in the q-exponential form. Taken together, our findings suggest that both mechanisms, copy process and preferential attachment, play a key role for the realization of networks with maximum Tsallis entropy. Finally, we discuss the interpretation of q parameter of the Tsallis entropy in the context of complex networks.     

Nuclear multifragmentation: Basic concepts

We present a brief overview of nuclear multifragmentation reaction. Basic formalism of canonical thermodynamical model based on equilibrium statistical mechanics is described. This model is used to calculate basic observables of nuclear multifragmentation like mass distribution, fragment multiplicity, isotopic distribution and isoscaling. Extension of canonical thermodynamical model to a projectile fragmentation model is outlined. Application of the projectile fragmentation model for calculating average number of intermediate mass fragments and the average size of the largest cluster at different Z _bound, differential charge distribution and cross-section of neutron-rich nuclei of different projectile fragmentation reactions at different energies are described. Application of nuclear multifragmentation reaction in basic research as well as in other domains is outlined.     

Onset of nuclear matter expansion in Au+Au collisions

Using the FOPI detector at GSI Darmstadt, excitation functions of collective flow components were measured for the Au+Au system, in the reaction plane and out of this plane, at seven incident energies ranging from 100A MeV to 800A MeV. The threshold energies, corresponding to the onset of sideward-flow (balance energy) and squeeze-out effect (transition energy), are extracted from extrapolations of these excitation functions toward lower beam energies for charged products with Z ⩾ 2. The transition energy is found to be larger than the balance energy. The impact parameter dependence of both balance and transition energies, when extrapolated to central collisions, suggests comparable although slightly higher values than the threshold energy for the radial flow. The relevant parameter seems to be the energy deposited into the system in order to overcome the attractive nuclear forces.     

Could saturation effects be visible in a future electron–ion collider?

We expect to observe parton saturation in a future electron–ion collider. In this Letter we discuss this expectation in more detail considering two different models which are in good agreement with the existing experimental data on nuclear structure functions. In particular, we study the predictions of saturation effects in electron–ion collisions at high energies, using a generalization for nuclear targets of the b-CGC model, which describes the ep HERA quite well. We estimate the total, longitudinal and charm structure functions in the dipole picture and compare them with the predictions obtained using collinear factorization and modern sets of nuclear parton distributions. Our results show that inclusive observables are not very useful in the search for saturation effects. In the small x region they are very difficult to disentangle from the predictions of the collinear approaches. This happens mainly because of the large uncertainties in the determination of the nuclear parton distribution functions. On the other hand, our results indicate that the contribution of diffractive processes to the total cross section is about 20% at large A   and small Q²$Q^2$, allowing for a detailed study of diffractive observables. The study of diffractive processes becomes essential to observe parton saturation.     

Calculation of excitation functions of the 5 4 , 5 6 , 5 7 , 5 8 Fe(p,n) reaction from threshold to 30 MeV

The cross-sections for the formation of ^54,56,57,58Co in the ^54,56,57,58Fe(p, n) reaction from threshold to 30 MeV protons have been theoretically calculated using the TALYS-1.4 nuclear model code, whereby we have studied major nuclear reaction mechanisms, including direct, pre-equilibrium and compound nuclear reaction. Subsequently, the level density and shell damping parameters have been adjusted and at the same time, the odd–even effects are well comprehended. The excitation functions have been compared with experimental nuclear data. It is observed that the theoretical cross-sections match fairly well. Proton-induced reaction cross-sections provide clues to understand the nuclear structure and offers a good testing ground for ideas about nuclear forces. In addition, complete information in this field is very much required for application in accelerator-driven subcritical system.     

Ab initio theory of complex electronic ground state of superconductors: I. Nonadiabatic modification of the Born-Oppenheimer approximation

The ARPES of high-T_c cuprates and theoretical results of low-Fermi energy band structure fluctuation for different groups of superconductors indicate that electron coupling to pertinent phonon modes drive system from adiabatic into anti-adiabatic state (ω>E_F). At these circumstances, not only Migdal-Eliashberg approximation is not valid, but basic adiabatic Born-Oppenheimer approximation (BOA) does not hold. At these circumstances, electronic structure has to be studied as explicitly dependent on instantaneous nuclear coordinates Q as well as on instantaneous nuclear momenta P.In the present paper—part I, it has been shown that Q, P-dependent modification of the BOA for ground electronic state can be derived by sequence of canonical transformations of the basis functions. The effect of nuclear coordinates and momenta on electronic structure is presented in the form of corrections to zero-, one- and two-particle terms of clamped nuclear Hamiltonian. In the anti-adiabatic state, correction to electronic ground state energy (zero-particle term correction) is negative and system can be stabilized in the anti-adiabatic state at distorted geometry with respect to adiabatic equilibrium structure and gap in one-particle spectrum of quasi-continuum states at Fermi level can be opened. Stabilization effect is solely the consequence of nuclear dynamics (P) that is crucial in anti-adiabatic state. It has been shown that nuclear dynamics also increases electron correlation until system at nuclear motion remains in a bound state. Corresponding corrections to electronic wave function are also specified.On the other hand, when system remains at vibration motion of nuclei in adiabatic state, the influence of nuclear dynamics (P-dependence) is negligible. In this case, all basic effects are covered through nuclear coordinates (Q-dependence) within the adiabatic BOA and standard results of solid-state (or molecular) physics are recovered.     

First excited 0+ states in the germanium isotopes via the Se(d, 6Li) reaction

A systematic study of the even-A germanium isotopes with mass 70 ≦ A ≦ 78 via the Se(d, ⁶Li) reaction has been performed at E_d = 45 MeV. The reaction products were momentum analysed and mass identified with a QMG/2 magnetic spectrograph and the accompanying focal-plane detector system. The main emphasis of this investigation was on the nature of the first excited 0⁺ states. The striking variation in strength from one isotope to the other already observed in the (p, t) reactions to the same final nuclei is also seen for the α-pickup reaction. Previously derived wave functions that assume the excited 0⁺₂ states to be pure proton configuration states can also account for the present results.     

高能诱发核反应中粒子产生的相对信息熵

在本实验中，引用了一个新的特征量，相对信息熵Ｒ，研究了能量为１４．６，６０，２００ＡＧｅＶ的氧离子和２００ＡＧｅＶ的硫离子诱发核反应的产生粒子多重数分布．在目前的能区中，Ｒ近似地能量无关．Ｒ对快度窗口大小依赖的饱和性，揭示了中心区的熵集中产生．实验结果与Ｌｕｎｄ模型的ＦＲＩＴＩＯＦ模拟计算的预言一致．     

Nucleon transfer in highly mass-asymmetric reaction systems between197au and relatively light projectiles in the energy region below 10 MeV/u

Mechanisms of heavy-ion induced nuclear reactions in mass-asymmetric systems were studied by focusing on the nucleon transfer reaction. Excitation functions and projected mean recoil ranges for the target-like products in³⁷-,¹⁶O-,¹⁴N-, and¹²C-induced reactions on¹⁹⁷Au were measured by means ofγ-ray spectrometry. The energy range studied was near the Coulomb barrier of the systems with incident energies below 10 MeV/u. Nucleon transfer reactions were discussed by distinguishing the products from quasi-elastic transfer (QET) and those from deep inelastic transfer (DIT). The tendency towards equilibration of theN/Z value and the energy damping, which are the characteristic features of DIT, were found in the production of Au isotopes. Observations are consistent with the generally accepted concept that QET takes place along a trajectory near the Coulomb trajectory. QET was made to be connected with the interaction radius and most of the cross section ratios were reproduced well by an extended tunneling model.     

Some features of the ionization of nuclear reaction products

The recoil-atom ionization of some isotopes from Sc to Po produced by heavy ioninduced nuclear reactions has been investigated. An analysis of the available data on the ionic charge distributions of nuclear reaction products has shown that atomic ionization in the mass range ofA~ 40–70 corresponds to the calculated values. It has been found that the contribution from additional ionization due to the formation of inner-shell vacancies as a result of the internal conversion of nuclear transitions grows as the atomic number and excitation energy of nuclei increase and reaches 90% for Po recoil atoms from fusion reactions.