The nuclear thermodynamic observables like the temperature, volume and the specific heat as obtained from isotopic ratios in hot disassembled nuclear matter are examined in the light of the S-matrix approach to the nuclear equation of state. The values of the observables, as extracted without inclusion of scattering effects are found to be modified appreciably in some cases when scattering between the fragment species is taken care of.
A self-consistent approach based on finite temperature Green's functions is used to investigate thermodynamic properties of nuclear matter. The internal energy is derived from the diagrams associated to the interaction energy. Pressure and entropy up to T=20 MeV are obtained from the generating functional form of the thermodynamic potential.
A new Fermionic Molecular Dynamics (FMD) model based on a Skyrme functional is proposed in this paper. After introducing the basic formalism, some first applications to nuclear structure and nuclear thermodynamics are presented
We investigate the relation between the binding energy and the Fermi energy and between different expressions for the pressure in cold nuclear matter. For a self-consistent calculation based on a $Φ$ derivable $T-$matrix approximation with off-shell propagators the thermodynamic relations are well satisfied unlike for a $G-$matrix or a $T-$matrix approach using quasi-particle propagators in the ladder diagrams.
In medium T-matrix calculations for symmetric nuclear matter at zero and finite temperatures are presented. The internal energy is calculated from the Galitskii-Koltun's sum rule and from the summation of the diagrams for the interaction energy. The pressure at finite temperature is obtained from the generating functional form of the thermodynamic potential. The entropy at high temperature is estimated and compared to expressions corresponding to a quasiparticle gas.
Both simple and sophisticated models are frequently used in an attempt to understand how real nuclei breakup when subjected to large excitation energies, a process known as nuclear multifragmentation. Many of these models assume equilibriumthermodynamics and produce results often interpreted as evidence of a phase transition. This work examines one class of models and employs standard thermodynamical procedure to explore the possible existence and nature of a phase transition. The role of various terms, e.g. Coulomb and surface energy, is discussed.
The density and temperature dependence of the nuclear symmetry free energy is investigated using microscopic two- and three-body nuclear potentials constructed from chiral effective field theory. The nuclear force models and many-body methods are benchmarked to properties of isospin-symmetric nuclear matter in the vicinity of the saturation density as well as the virial expansion of the neutron matter equation of state at low fugacities. The free energy per particle of isospin-asymmetric nuclear matter is calculated assuming a quadratic dependence of the interaction contributions on the isospi…
We investigate the thermodynamic equation of state of isospin-symmetric nuclear matter with microscopic nuclear forces derived within the framework of chiral effective field theory. Two- and three-body nuclear interactions constructed at low resolution scales form the basis for a perturbative calculation of the finite-temperature equation of state. The nuclear force models and many-body methods are benchmarked against bulk properties of isospin-symmetric nuclear matter at zero temperature, which are found to be well reproduced when chiral nuclear interactions constructed at the lowest resoluti…
We calculate thermodynamic quantities in symmetric nuclear matter within the self-consistent Green's functions method including three-body forces. The thermodynamic potential is computed directly from a diagrammatic expansion, implemented with the CD-Bonn and Nijmegen nucleon-nucleon potentials and the Urbana three-body forces. We present results for entropy and pressure up to temperatures of 20 MeV and densities of 0.32 fm^-3. While the pressure is sensitive to the inclusion of three-body forces, the entropy is not. The unstable spinodal region is identified and the critical temperature assoc…
Direct nuclear reactions $\vec{d}+A$ and $\vec{p}+(An)$ are described in the framework of three-body Faddeev-type equations. Differential cross section and analyzing powers are calculated using several optical potential models and compared with the experimental data. Quite satisfactory agreement is found except for few systematic discrepancies.
The temperature dependence of the chiral condensate in isospin-symmetric nuclear matter at varying baryon density is investigated using thermal in-medium chiral effective field theory. This framework provides a realistic approach to the thermodynamics of the correlated nuclear many-body system and permits calculating systematically the pion-mass dependence of the free energy per particle. One- and two-pion exchange processes, $Δ(1232)$-isobar excitations, Pauli blocking corrections and three-body correlations are treated up to and including three loops in the expansion of the free energy densi…
We investigate properties of nuclear matter and examine possible correlations with neutron star observables for a set of microscopic nuclear equations of state derived within the Brueckner-Hartree-Fock formalism employing compatible three-body forces. We find good candidates for a realistic nuclear EOS up to high density and confirm strong correlations between neutron star radius, tidal deformability, and the pressure of betastable matter. No correlations are found with the saturation properties of nuclear matter.
We explore the use of mean field models to approximate microscopic nuclear equations of state derived from chiral effective field theory across the densities and temperatures relevant for simu- lating astrophysical phenomena such as core-collapse supernovae and binary neutron star mergers. We consider both relativistic mean field theory with scalar and vector meson exchange as well as energy density functionals based on Skyrme phenomenology and compare to thermodynamic equa- tions of state derived from chiral two- and three-nucleon forces in many-body perturbation theory. Quantum Monte Carlo s…
The computation of the thermodynamic properties of nuclear matter is a central task of theoretical nuclear physics. The nuclear equation of state is an essential quantity in nuclear astrophysics and governs the properties of neutron stars and core-collapse supernovæ. The framework of chiral effective field theory provides the basis for the description of nuclear interactions in terms of a systematic low-energy expansion. In this thesis, we apply chiral two- and three-nucleon interactions in perturbative many-body calculations of the thermodynamic equation of state of infinite homogeneous nucle…
Meaningful predictions for electric quadrupole (E2) observables from ab initio nuclear theory are necessary, if the ab initio description of collective correlations is to be confronted with experiment, as well as to provide predictive power for unknown E2 observables. However, converged results for E2 observables are notoriously challenging to obtain in ab initio no-core configuration interaction (NCCI) approaches. Matrix elements of the E2 operator are sensitive to the large-distance tails of the nuclear wave function, which converge slowly in an oscillator basis expansion. Similar convergenc…
Different formulas relying measurable fragment isotopic observables to the symmetry energy of excited nuclei have been proposed and applied to the analysis of heavy ion collision data in the recent literature. In this paper we examine the quality of the different expressions in the framework of the McGill Canonical Thermodynamic Model. We show that even in the idealized situation of canonical equilibrium and in the absence of secondary decay, these formulas do not give a precise reconstruction of the symmetry energy of the fragmenting source. However, both isotopic widths and isoscaling appear…
On the basis of the kinetic equation with selfconsistent relativistic mean fields acting on baryons and antibaryons, we study dynamical response of the nucleus to an antiproton implanted in its interior. By solving numerically the time-dependent Vlasov equation, we show that the compressed state is formed on a rather short time scale of about 4-10 fm/c. This justifies the assumption, that the antiproton annihilation may happen in the compressed nuclear environment. The evolution of the nucleus after antiproton annihilation is described by the same kinetic equation including collision terms. We…
The thermal evolution of a few thermodynamic properties of the nuclear surface like its thermodynamic potential energy, entropy and the symmetry free energy are examined for both semi-infinite nuclear matter and finite nuclei. The Thomas-Fermi model is employed. Three Skyrme interactions, namely, SkM$^*$, SLy4 and SK255 are used for the calculations to gauge the dependence of the nuclear surface properties on the energy density functionals. For finite nuclei, the surface observables are computed from a global liquid-drop inspired fit of the energies and free energies of a host of nuclei coveri…
Properties of nuclear and neutron matter are discussed in a nonlinear $σ$-$ω$-$ρ$ mean-field approximation with self-interactions and mixing-interactions of mesons and baryons. The nonlinear interactions are renormalized by employing the theory of conserving approximations, which results in a thermodynamically consistent approximation that maintains Hugenholtz-Van Hove theorem and Landau's requirement of quasiparticles. The approximation is equivalent to the Hartree approximation with {\it effective masses} and {\it effective coupling constants} of baryons and mesons. The effective masses and …
We explore the presence of thermodynamic instabilities and, con\-se\-quen\-tly, the realization of a pure hadronic phase transition in the hot and finite baryon density nuclear matter. The analysis is performed by means of an effective relativistic mean-field model with the inclusion of hyperons, $Δ$-isobars, and the lightest pseudoscalar and vector meson degrees of freedom. The Gibbs conditions on the global conservation of baryon number and zero net strangeness in symmetric nuclear matter are required. Similarly to the liquid-gas phase transition, we show that a phase transition, characteriz…
Two-particle correlations are a widely used tool for studying relativistic nuclear collisions. Multiplicity fluctuations comparing charge and particle species have been studied as a possible signal for Quark-Gluon Plasma (QGP) and the QCD critical point. These fluctuation studies all make use of particle variances which can be shown to originate with a two-particle correlation function. Momentum correlations and covariances of momentum fluctuations, which arise from the same correlation function, have also been used to extract properties of the nuclear collision medium such as the shear viscos…
The free energy and the equation of state of isospin-asymmetric nuclear matter are calculated at finite temperature up to three loop order in the framework of in-medium chiral perturbation theory, systematically incorporating one- and two-pion exchange dynamics to this order. Effects from the two-pion exchange with explicit Δ-isobar excitation are included, as well as three-body forces. We construct the phase diagram of nuclear matter for different proton fractions x_p and investigate the dependence of nuclear matter properties on the isospin-asymmetry. A detailed study of the liquid-gas phase…
This presentation reviews an approach to nuclear many-body systems based on the spontaneously broken chiral symmetry of low-energy QCD. In the low-energy limit, for energies and momenta small compared to a characteristic symmetry breaking scale of order 1 GeV, QCD is realized as an effective field theory of Goldstone bosons (pions) coupled to heavy fermionic sources (nucleons). Nuclear forces at long and intermediate distance scales result from a systematic hierarchy of one- and two-pion exchange processes in combination with Pauli blocking effects in the nuclear medium. Short distance dynamic…
The behaviour of a di-nuclear system in the regime of strong pairing correlations is studied with the methods of statistical mechanics. It is shown that the thermal averaging is strong enough to assure the application of thermodynamical methods to the energy exchange between the two nuclei in contact. In particular, thermal averaging justifies the definition of a nuclear temperature.
We address two questions pertaining to the uniqueness and usefulness of a new observable: (i) Considering the current theoretical knowledge, what novel information does new measurement bring in? (ii) How can new data reduce uncertainties of current theoretical models? We illustrate these points by studying the radius of the neutron distribution of a heavy nucleus, a quantity related to the equation of state for neutron matter that determines properties of nuclei and neutron stars. By systematically varying parameters of two theoretical models and studying the resulting confidence ellipsoid, we…
We calculate the equation of state of nuclear matter for arbitrary isospin-asymmetry up to three loop order in the free energy density in the framework of in-medium chiral perturbation theory. In our approach 1π- and 2π-exchange dynamics with the inclusion of the Δ-isobar excitation as an explicit degree of freedom, corresponding to the long- and intermediate-range correlations, are treated explicitly. Few contact terms fixed to reproduce selected known properties of nuclear matter encode the short-distance physics. Two-body as well as three-body forces are systematically included. We find a c…
Starting from a set of different two- and three-nucleon interactions from chiral effective field theory, we use the importance-truncated no-core shell model for ab initio calculations of excitation energies as well as electric quadrupole (E2) and magnetic dipole (M1) moments and transition strengths for selected p-shell nuclei. We explore the sensitivity of the excitation energies to the chiral interactions as a first step towards and systematic uncertainty propagation from chiral inputs to nuclear structure observables. The uncertainty band spanned by the different chiral interactions is typi…
The BCS-BEC crossover and phase diagram for asymmetric nuclear superfluid with pairings in isospin I = 0 and I = 1 channels are investigated at mean field level, by using a density dependent nucleon-nucleon potential. Induced by the in-medium nucleon mass and density dependent coupling constants, neutron-proton Cooper pairs could be in BEC state at sufficiently low density, but there is no chance for the BEC formation of neutron-neutron and proton-proton pairs at any density and asymmetry. We calculate the phase diagram in asymmetry-temperature plane for weakly interacting nuclear superfluid, …
The temperature dependence of the chiral condensate in isospin-symmetric nuclear matter at varying baryon densities is investigated starting from a realistic free energy density of the correlated nuclear many-body system. The framework is thermal in-medium chiral effective field theory which permits a systematic calculation of the quark mass dependence of the free energy density. One- and two-pion exchange processes, virtual $Δ(1232)$-isobar excitations and Pauli blocking corrections are treated up to and including three-loop order. It is found that nuclear matter remains in the Nambu-Goldston…
We develop a model in the framework of nuclear fragmentation at thermodynamic equilibrium which can be mapped onto an Ising model with constant magnetization. We work out the thermodynamic properties of the model as well as the properties of the fragment size distributions. We show that two types of phase transitions can be found for high density systems. They merge into a unique transition at low density. An analysis of the critical exponents which characterize observables for different densities in the thermodynamic limit shows that these transitions look like continuous second order transit…
We explore the conditions under which the particle number conservation constraint deforms the predictions of fragmentation observables as calculated in the grand canonical ensemble. We derive an analytical formula allowing to extract canonical results from a grand canonical calculation and vice versa. This formula shows that exact canonical results can be recovered for observables varying linearly or quadratically with the number of particles, independent of the grand canonical particle number fluctuations. We explore the validity of such grand canonical extrapolation for different fragmentati…
Equilibrium statistics of finite Hamiltonian systems is fundamentally described by the microcanonical ensemble (ME). Canonical, or grand-canonical partition functions are deduced from this by Laplace transform. Only in the thermodynamic limit are they equivalent to ME for homogeneous systems. Therefore ME is the only ensemble for non-extensive/inhomogeneous systems like nuclei or stars where the $\lim_{N\to \infty,ρ=N/V=const}$ does not exist. Conventional canonical thermo-statistic is inapplicable for non-extensive systems. This has far reaching fundamental and quite counter-intuitive consequ…
The appearance and disappearance of shells and subshells are determined using a previously introduced method of structural analysis. This work extends the approach and applies it to protons, in addition to neutrons, in an attempt to provide a more complete understanding of shell structure in nuclei. Experimental observables including the mean-square charge radius, as well as other spectroscopic and mass related quantities are analyzed for extrema. This analysis also uses differential observables among adjacent even-even nuclei to serve as the derivatives for these quantities of interest. Local…
We study the neutron-proton pairing in nuclear matter as a function of isospin asymmetry at finite temperatures and the saturation density using realistic nuclear forces and Brueckner-renormalized single particle spectra. Our computation of the thermodynamic quantities shows that while the difference of the entropies of the superconducting and normal phases anomalously changes its sign as a function of temperature for arbitrary asymmetry, the grand canonical potential does not; the superconducting state is found to be stable in the whole temperature-asymmetry plane. The pairing gap completely …
The strictly reversible, thermodynamically equilibrium nature of the free rotation of a body makes it possible to obtain a number of bounds on the rotational characteristics within individual rotational bands of nonspherical nuclei. As a result, the bounds between which the possible values of the critical spin $J_c$ lie can be expressed exclusively in terms of a restricted number of the experimentally most accessible data on the lower phase $J<J_c$ for a given nuclide. The bounds are tested on the ground-state rotational bands (yrast lines) of even-even nuclei, in which the corresponding phase…
Nuclear thermodynamics studies evidenced the existence of a first order phase transition, namely of the liquid-gas type, without paying attention to the isospin degree of freedom. On the other hand, only few results with the introduction of the isospin variable have been so far obtained. Moreover above all a key question remains. It concerns the origin of the dynamics of the phase transition: spinodal instabilities or not with possible consequences and new signatures related to the introduction of the isospin variable.
Critical behaviors of quark-hadron phase transition in high-energy heavy-ion collisions are investigated with the aim of identifying hadronic observables. The surface of the plasma cylinder is mapped onto a 2D lattice. The Ising model is used to simulate configurations corresponding to cross-over transitions in accordance to the findings of QCD lattice gauge theory. Hadrons are formed in clusters of all sizes. Various measures are examined to quantify the fluctuations of the cluster sizes and of the voids among the clusters. The canonical power-law behaviors near the critical temperature are f…
An exact analytical solution of the statistical multifragmentation model is found in thermodynamic limit. The model exhibits a 1-st order phase transition of the liquid-gas type. The mixed phase region of the phase diagram, where the gas of nuclear fragments coexists with the infinite liquid condensate, is unambiguously identified. The peculiar thermodynamic properties of the model near the boundary between the mixed phase and the pure gaseous phase are studied. The results for the caloric curve and specific heat are presented and a physical picture of the nuclear liquid-gas phase transition i…
Progress on nuclear liquid gas phase transition (LGPT) or critical behavior has been simply reviewed and some signals of LGPT in heavy ion collisions, especially in NIMROD data, are focused. These signals include the power-law charge distribution, the largest fluctuation of the fragment observables, the nuclear Zipf law, caloric curve and critical exponent analysis etc.
One of the most fundamental questions in the field of relativistic heavy ion physics is how to reach and explore densities which are needed to cross the chiral and/or the deconfinement phase transition. In this analysis we investigate the information we can gather by analyzing baryonic and mesonic resonances on the hot and dense phase in such nuclear reactions. The decay products of these resonances carry information on the resonances properties at the space time point of their decay. We especially investigate the percentage of reconstructable resonances as a function of density for heavy ion …
Analytical formulae are presented which provide quantitative estimates for the suppression of the anticipated back-to-back particle--antiparticle correlations in high energy nuclear collisions due to the finite duration of the transition dynamics. They show that it is unlikely to observ the effect.
The dynamical description of correlated nuclear motion is based on a set of coupled equations of motion for the one-body density matrix $ρ(11';t)$ and the two-body correlation function $c_2(12,1'2';t)$, which is obtained from the density-matrix hierarchy beyond conventional mean-field approaches by truncating 3-body correlations. The resulting equations nonperturbatively describe particle-particle collisions (short-range correlations) as well as particle-hole interactions (long-range correlations). Within a basis of time-dependent Hartree-Fock states these equations of motion are solved for co…
Calculations of nuclei are often carried out in finite model spaces. Thus, finite-size corrections enter, and it is necessary to extrapolate the computed observables to infinite model spaces. In this work, we employ extrapolation methods based on artificial neural networks for observables such as the ground-state energy and the point-proton radius. We extrapolate results from no-core shell model and coupled-cluster calculations to very large model spaces and estimate uncertainties. Training the network on different data typically yields extrapolation results that cluster around distinct values…
We study symmetric nuclear matter at finite temperature, with particular emphasis on the liquid-gas phase transition. We use a standard covariance analysis to propagate statistical uncertainties from the density functional to the thermodynamic properties. We use four functionals with known covariance matrices to obtain as wide a set of results as possible. Our findings suggest that thermodynamical properties are very well constrained by fitting data at zero temperature. The propagated statistical errors in the liquid-gas phase transition parameters are relatively small.
We present the quantum computation of nuclear observables where the operators of interest are first decomposed in terms of the linear combination of unitaries. Then we utilise the Hadamard test and the linear combination of unitaries (LCU) based methods to compute the expectation values. We apply these methods to calculate the electric quadrupole moment of deuteron. The results are compared for the Jordan-Wigner transformation and Gray code encoding. We discuss the versatility of our approach that can be utilized in general to calculate several observables on a quantum computer.
We test the compatibility of chiral symmetry, dynamic mass generation of the nucleon due to spontaneous breaking of chiral symmetry, and the description of finite nuclear systems by employing an NJL model understood as a chiral invariant effective theory for nucleons. We apply the model to nuclear matter as well as to finite nuclei. In the latter case, the model is adjusted to nuclear ground-state observables. We treat the case of a pure chiral theory and the physically more realistic case where a portion of the nucleon mass (160 MeV) explicitly breaks chiral symmetry. The best version of this…
We present results obtained in the calculation of nuclear ground state properties in relativistic Hartree approximation using a Lagrangian whose QCD-scaled coupling constants are all natural (dimensionless and of order 1). Our model consists of four-, six-, and eight-fermion point couplings (contact interactions) together with derivative terms representing, respectively, two-, three-, and four-body forces and the finite ranges of the corresponding mesonic interactions. The coupling constants have been determined in a self-consistent procedure that solves the model equations for representative …
The prediction of nuclear observables beyond the finite model spaces that are accessible through modern ab initio methods, such as the no-core shell model, pose a challenging task in nuclear structure theory. It requires reliable tools for the extrapolation of observables to infinite many-body Hilbert spaces along with reliable uncertainty estimates. In this work we present a universal machine learning tool capable of capturing observable-specific convergence patterns independent of nucleus and interaction. We show that, once trained on few-body systems, artificial neural networks can produce …
Within the relativistic PWIA, spin observables have been recalculated for quasielastic ($\vec p,\vec p '$) and ($\vec p,\vec n$) reactions on a $^{40}$Ca target. The incident proton energy ranges from 135 to 300 MeV while the transferred momentum is kept fixed at 1.97 fm^{-1}. In the present calculations, new Horowitz-Love--Franey relativistic NN amplitudes have been generated in order to yield improved and more quantitative spin observable values than before. The sensitivities of the various spin observables to the NN interaction parameters, such as (1) the presence of the surrounding nuclear…
The standard approach to nuclear physics encodes phase shift information in an NN potential, then decodes that information in forming an effective interaction, appropriate to a low-momentum Hilbert space. Here we show that it is instead possible to construct the effective interaction directly from continuum phase shifts and mixing angles, eliminating all reference to a high momentum potential. The theory is rapidly convergent and well behaved, yielding sub-keV accuracy.
We describe a finite inhomogeneous three dimensional system of classical particles which interact through short and (or) long range interactions by means of a simple analytic spin model. The thermodynamic properties of the system are worked out in the framework of the grand canonical ensemble. It is shown that the system experiences a phase transition at fixed average density in the thermodynamic limit. The phase diagram and the caloric curve are constructed and compared with numerical simulations. The implications of our results concerning the caloric curve are discussed in connection with th…
The nuclear symmetry energy represents a response to the neutron-proton asymmetry. In this survey we discuss various aspects of symmetry energy in the framework of nuclear density functional theory, considering both non-relativistic and relativistic self-consistent mean-field realizations side-by-side. Key observables pertaining to bulk nucleonic matter and finite nuclei are reviewed. Constraints on the symmetry energy and correlations between observables and symmetry-energy parameters, using statistical covariance analysis, are investigated. Perspectives for future work are outlined in the co…
We study thermodynamics of nuclear matter in a two-flavored parity doublet model within the mean field approximation. Parameters of the model are chosen to reproduce correctly the properties of the nuclear ground state. The model predicts two phase transitions in nuclear matter, a liquid-gas phase transition at normal nuclear density and a chiral transition at higher density. At finite temperature the pion decay constant exhibits a considerable reduction at intermediate values of chemical potential, which is traced back to the presence of the liquid-gas transition, and approaches zero at highe…
Spin observables in radiative neutron capture on a proton and its inverse process, photodisintegration of the deuteron are calculated using a pionless effective field theory with di-baryon fields. Good agreement with the results of existing standard nuclear physics approach is obtained at very low energies. As energy increases, however, the discrepancy between the effective field theory and the standard nuclear physics approach becomes substantial. We discuss the origin of the difference.
We discuss how the possible changes of hadron properties can be tested, associated with the chiral transition in nuclear medium; the hadrons discussed are the vector mesons,the $σ$ meson and the chiral partner of the nucleon. We emphasize that the proper quantity is the spectral function which describes the nuclear medium having a hadron; the analysis of electro-quasielastic reactions, for instance, has a relevance to the physics of CERES. The similarity is indicated between the physics discussed here with that of giant resonances. Experiments using nuclear targets are proposed to observe the …
Using a pQCD picture for jet evolution in a dense QCD medium, in which medium-induced parton branchings are factorized from vacuum-like emissions, we study two jet substructure observables: the $z_g$ and the Soft Drop multiplicity distributions. We compute the respective nuclear modification factors using a Monte-Carlo implementation of the parton showers. Our results are in qualitative agreement with LHC data for Pb+Pb collisions. We identify the physical mechanisms explaining our results: incoherent jet energy loss, semi-hard medium-induced emissions and the bias introduced by the steeply fa…
The traditional approach to nuclear physics encodes phase shift information in a nucleon-nucleon (NN) potential, producing a nucleon-level interaction that captures the sub-GeV consequences of QCD. A further reduction to the nuclear scale is needed to produce an effective interaction for soft Hilbert spaces, such as those employed in the shell model. Here we describe an alternative construction of this effective interaction, from QCD directly to the nuclear scale, that is direct and precise. This eliminates the need for constructing and renormalizing the high-momentum NN potential. Instead, co…
We propose a new Monte Carlo method called the pinhole trace algorithm for {\it ab initio} calculations of the thermodynamics of nuclear systems. For typical simulations of interest, the computational speedup relative to conventional grand-canonical ensemble calculations can be as large as a factor of one thousand. Using a leading-order effective interaction that reproduces the properties of many atomic nuclei and neutron matter to a few percent accuracy, we determine the location of the critical point and the liquid-vapor coexistence line for symmetric nuclear matter with equal numbers of pro…
