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Strong mean meson fields, which are known to exist in normal nuclei, experience a violent deformation in the course of a heavy-ion collision at relativistic energies. This may give rise to a new collective mechanism of the particle production, not reducible to the superposition of elementary nucleon-nucleon collisions.
We investigate the sensivity of pionic bounce-off and squeeze-out on the density and momentum dependence of the real part of the nucleon optical potential. For the in-plane pion bounce-off we find a strong sensivity on both the density and momentum dependence whereas the out-of-plane pion squeeze-out shows a strong sensivity only towards the momentum dependence but little sensivity towards the density dependence.
We demonstrate the importance of the Bose-statistical effects for pion production in relativistic heavy-ion collisions. The evolution of the pion phase-space density in central collisions of ultrarelativistic nuclei is studied in a simple kinetic model taking into account the effect of Bose-simulated pion production by the NN collisions in a dense cloud of mesons.
We demonstrate that the creation of strange matter is conceivable in the midrapidity region of heavy ion collisions at Brookhaven RHIC and CERN LHC. A finite net-baryon density, abundant (anti)strangeness production, as well as strong net-baryon and net-strangeness fluctuations, provide suitable initial conditions for the formation of strangelets or metastable exotic multistrange ( baryonic) objects. Even at very high initial entropy per baryon SyAinit ¯ 500 and low initial baryon numbers of Ainit B ¯ 30 a quark-gluon-plasma droplet can immediately charge up with strangeness and accumulate net-baryon number. PACS numbers: 25.75.Dw, 12.38.Mh, 24.85.+
Measured hadron yields from relativistic nuclear collisions can be equally well understood in two physically distinct models, namely a static thermal hadronic source versus a time-dependent, non-equilibrium hadronization off a quark gluon plasma droplet. Due to the time-dependent particle evaporation off the hadronic surface in the latter approach the hadron ratios change (by factors of / 5) in time. The overall particle yields then reflect time averages over the actual thermodynamic properties of the system at a certain stage of evolution.
Abstract: Local thermal and chemical equilibration is studied for central AqA collisions at 10.7 160 AGeV in the Ultrarelativis- . tic Quantum Molecular Dynamics model UrQMD . The UrQMD model exhibits strong deviations from local equilibrium at the high density hadron string phase formed during the early stage of the collision. Equilibration of the hadron resonance matter is established in the central cell of volume Vs125 fm3 at later stages, tG10 fmrc, of the resulting quasi-isentropic expansion. The thermodynamical functions in the cell and their time evolution are presented. Deviations of the UrQMD quasi-equilibrium state from the statistical mechanics equilibrium are found. They increase with energy per baryon and lead to a strong enhancement of the pion number density as compared to statistical mechanics estimates at SPS energies. PACS: 25.75.-q; 24.10.Lx; 24.10.Pa; 64.30.qt
Noneequilibrium models (three-fluid hydrodynamics and UrQMD) use to discuss the uniqueness of often proposed experimental signatures for quark matter formation in relativistic heavy ion collisions. It is demonstrated that these two models - although they do treat the most interesting early phase of the collisions quite differently(thermalizing QGP vs. coherent color fields with virtual particles) - both yields a reasonable agreement with a large variety of the available heavy ion data.
We study J/psi suppression in AB collisions assuming that the charmonium states evolve from small, color transparent configurations. Their interaction with nucleons and nonequilibrated, secondary hadrons is simulated using the microscopic model UrQMD. The Drell-Yan lepton pair yield and the J/psi Drell-Yan ratio are calculated as a function of the neutral transverse energy in Pb+Pb collisions at 160 GeV and found to be in reasonable agreement with existing data.
We derive the relativistic quantum transport equation for the pion distribution function based on an effective Lagrangian of the QHD-II model. The closed-time-path Green s function technique and the semiclassical, quasiparticle, and Born approximations are employed in the derivation. Both the mean field and collision term are derived from the same Lagrangian and presented analytically. The dynamical equation for the pions is consistent with that for the nucleons and Delta's which we developed before. Thus, we obtain a relativistic transport model which describes the hadronic matter with N,Delta, and pi degrees of freedom simultaneously. Within this approach, we investigate the medium effects on the pion dispersion relation as well as the pion absorption and pion production channels in cold nuclear matter. In contrast to the results of the nonrelativistic model, the pion dispersion relation becomes harder at low momenta and softer at high momenta as compared to the free one, which is mainly caused by the relativistic kinetics. The theoretically predicted free piN->Delta cross section is in agreement with the experimental data. Medium effects on the piN->Delta cross section and momentum-dependent Delta-decay width are shown to be substantial. PACS-numbers: 24.10.Jv, 13.75.Cs, 21.65.1f, 25.75.2q
We calculate the shadowing of sea quarks and gluons and show that the shadowing of gluons is not simply given by the sea quark shadowing, especially at small x. The calculations are done in the lab frame approach by using the generalized vector meson dominance model. Here the virtual photon turns into a hadronic fluctuation long before the nucleus. The subsequent coherent interaction with more than one nucleon in the nucleus leads to the depletion sigma(gamma*A )< A*sigma(gamma * N) known as shadowing. A comparison of the shadowing of quarks to E665 data for 40Ca and 207Pb shows good agreement.
A generic property of a first-order phase transition in equilibrium, and in the limit of large entropy per unit of conserved charge, is the smallness of the isentropic speed of sound in the mixed phase . A specific prediction is that this should lead to a non-isotropic momentum distribution of nucleons in the reaction plane (for energies < 40A GeV in our model calculation). On the other hand, we show that from present effective theories for low-energy QCD one does not expect the thermal transition rate between various states of the effective potential to be much larger than the expansion rate, questioning the applicability of the idealized Maxwell/Gibbs construction. Experimental data could soon provide essential information on the dynamics of the phase transition.
The lightest supersymmetric particle, most likely the neutralino, might account for a large fraction of dark matter in the Universe.We show that the primordial spectrum of density fluctuations in neutralino cold dark matter (CDM) has a sharp cut-off due to two damping mechanisms: collisional damping during the kinetic decoupling of the neutralinos at (10 MeV) and free streaming after last scattering of neutralinos. The cut-off in the primordial spectrum defines a minimal mass for CDM objects in hierarchical structure formation. For typical neutralino and sfermion masses the first gravitionally bound neutralino clouds have masses above 10 -6 M .
We study the bound states of anti-nucleons emerging from the lower continuum in finite nuclei within the relativistic Hartree approach including the contributions of the Dirac sea to the source terms of the meson fields. The Dirac equation is reduced to two Schr¨odinger-equivalent equations for the nucleon and the anti-nucleon respectively. These two equations are solved simultaneously in an iteration procedure. Numerical results show that the bound levels of anti-nucleons vary drastically when the vacuum contributions are taken into account. PACS number(s): 21.10.-k; 21.60.-n; 03.65.Pm
Recent progress in the understanding of the high density phase of neutron stars advances the view that a substantial fraction of the matter consists of hyperons. The possible impacts of a highly attractive interaction between hyperons on the properties of compact stars are investigated.We find that a hadronic equation of state with hyperons allows for a first order phase transition to hyperonic matter. The corresponding hyperon stars can have rather small radii of R 8 km.
The production of black holes at Tevatron and LHC in spacetimes with compactified space-like large extra dimensions is studied. Either black holes can already be observed in ¯ pp collisions at s = 1.8 TeV or the fundamental gravity scale has to be above 1.4 TeV. At LHC the creation of a large number of quasi-stable black holes is predicted, with lifetimes beyond several hundred fm/c. A cut-off in the high-PT jet cross section is shown to be a unique signature of black hole production. This signal is compared to the jet plus missing energy signature due to graviton production in the final state as proposed by the ATLAS collaboration.
Abstract Geant4 is a toolkit for simulating the passage of particles through matter. It includes a complete range of functionality including tracking, geometry, physics models and hits. The physics processes offered cover a comprehensive range, including electromagnetic, hadronic and optical processes, a large set of long-lived particles, materials and elements, over a wide energy range starting, in some cases, from 250 eV and extending in others to the TeV energy range. It has been designed and constructed to expose the physics models utilised, to handle complex geometries, and to enable its easy adaptation for optimal use in different sets of applications. The toolkit is the result of a worldwide collaboration of physicists and software engineers. It has been created exploiting software engineering and object-oriented technology and implemented in the C++ programming language. It has been used in applications in particle physics, nuclear physics, accelerator design, space engineering and medical physics. PACS: 07.05.Tp; 13; 23
The amount of proton stopping in central Pb+Pb collisions from 20–160 A GeV as well as hyperon and antihyperon rapidity distributions are calculated within the UrQMD model in comparison to experimental data at 40, 80, and 160 A GeV taken recently from the NA49 collaboration. Furthermore, the amount of baryon stopping at 160A GeV for Pb+Pb collisions is studied as a function of centrality in comparison to the NA49 data. We find that the strange baryon yield is reasonably described for central collisions, however, the rapidity distributions are somewhat more narrow than the data. Moreover, the experimental antihyperon rapidity distributions at 40, 80, and 160 A GeV are underestimated by up to factors of 3—depending on the annihilation cross section employed—which might be addressed to missing multimeson fusion channels in the UrQMD model. Pacs-Nr.: 25.75.2q, 24.10.Jv, 24.10.Lx
Accelerating cavities exchange HOM power through interconnecting beam pipes in case of signal frequencies above the cut-off of their propagating waveguide modes. This may lead either to improved HOM damping or - in the case most severe - to unwanted phase coherence of fields to the beam. Therefore the knowledge of the scattering properties of a cavity as a line element is needed to analyse all kinds of RF cavity-cavity interaction. Since there is a lack of measurement tools capable to provide a multidimensional scattering matrix at a given frequency point, we have been developing a method for this purpose. It uses a set of 2-port S-parameters of the device under test, embedded in a number of geometrically different RF environments. The application of the method is demonstrated with copper models of TESLA cavities.
Abstract: The measured particle ratios in central heavy-ion collisions at RHIC-BNL are investigated within a chemical and thermal equilibrium chiral SU(3) Ã É approach. The commonly adopted non-interacting gas calculations yield temperatures close to or above the critical temperature for the chiral phase transition, but without taking into account any interactions. In contrast, the chiral SU(3) model predicts temperature and density dependent effective hadron masses and effective chemical potentials in the medium and a transition to a chirally restored phase at high temperatures or chemical potentials. Three different parametrizations of the model, which show different types of phase transition behaviour, are investigated. We show that if a chiral phase transition occured in those collisions, freezing of the relative hadron abundances in the symmetric phase is excluded by the data. Therefore, either very rapid chemical equilibration must occur in the broken phase, or the measured hadron ratios are the outcome of the dynamical symmetry breaking. Furthermore, the extracted chemical freeze-out parameters differ considerably from those obtained in simple non-interacting gas calculations. In particular, the three models yield up to 35 MeV lower temperatures than the free gas approximation. The inmedium masses turn out to differ up to 150 MeV from their vacuum values.
By replacing the irises in an electron linac by a slit one gets a structure capable of focussing/defocussing an electron beam (rf-quadrupoles). Therefore one can think of a combination of rf- and conventional magnetic quadrupoles for transversal focussing in linear-colliders. Furthermore they can meet the demands of BNS-damping without initial energy spread. Considering multibunch-operation of a collider, the long-range wake behaviour of this kind of structure has to be investigated. A three-cell structure has been built and investigated for dipole-type transversal long-range wakes. The experimental results are compared to numerical simulations done with MAFIA.
The effect of a single HOM-damper cell within a channel of undamped cells is described theoretically using an equivalent circuit model. From this a simple equation can be derived which relates the Q-value of the single damping-cell, the bandwidth of the passband under consideration, and the additional phase shift which is introduced by the damper cell to provide energy flow into the damper cell. This equation immediately shows the limitations of such single cell damping systems. Comparisons with experimental results are shown.
A new method for measuring quality factors in cavities is presented. This method is capable of measuring Q-factors in heavily damped as well as in undamped cavities. In addition, the possibility of separating overlapping modes and measuring their Q-factors is provided. Measurements on HOM (higher order mode) damped cavities for the DESY/THD linear collider project are presented
We study the collective flow of open charm mesons and charmonia in Au + Au collisions at s = 200 GeV within the hadron-string-dynamics (HSD) transport approach. The detailed studies show that the coupling of D, mesons to the light hadrons leads to comparable directed and elliptic flow as for the light mesons. This also holds approximately for J/ mesons since more than 50% of the final charmonia for central and midcentral collisions stem from D + induced reactions in the transport calculations. The transverse momentum spectra of D, mesons and J/ s are only very moderately changed by the (pre-)hadronic interactions in HSD, which can be traced back to the collective flow generated by elastic interactions with the light hadrons. PACS-Nr. 25.75.-q, 13.60.Le, 14.40.Lb, 14.65.Dw
Mode propagation in an iris type accelerator section loaded with single heavily HOM-damped cells
(1994)
The wakefield effects in accelerator sections for future linear colliders will be reduced either by damping by detuning or by a combination of both. For the DESY/THD linac [1] it is forseen to employ heavily HOM-damped cells to provide a strong coupling to the TE/TM11-dipole passband as well as to the TM/TE11-dipole passband. For our experiments we have used wall-slotted damping cells. This leads to several problems concerning the propagation of fundamental and HOM-modes. Experimental investigations have been done. Results are presented.
The operation of a Free Electron Laser (FEL) in the ultraviolet or in the X-ray regime requires the acceleration of electron bunches with an rms length of 25 to 50 mikrometer. The wakefields generated by these sub picosecond bunches extend into the frequency range well beyond the threshold for Cooper pair breakup (about 750 GHz) in superconducting niobium at 2 K. It is shown, that the superconducting cavities can indeed be operated with 25 mikrometer bunches without suffering a breakdown of superconductivity (quench), however at the price of a reduced quality factor and an increased heat transfer to the superfluid helium bath. This was first shown by wakefield calculations based on the diffraction model [1]. In the meantime a more conventional method of computing wake fields in the time domain by numerical methods was developed and used for the wakefield calculations [2]. Both methods lead to comparable results: the operation of TESLA with 25 mikrometer bunches is possible but leads to an additional heat load due to the higher order modes (HOMs). Therefore HOM dampers for these high frequencies are under construction [3]. These dampers are located in the beam pipes between the 9-cell cavities. So it is of interest, if there are trapped modes in the cavity due to closed photon orbits. In this paper we investigate the existence of trapped modes and the distribution of heat load over the surface of the TESLA cavity by numerical photon tracking.
Determination of field strength and quality factor of heavily HOM damped accelerator cavities
(1992)
Two methods of of measuring field strength in accelerator cavities, heavily damped with respect to higher order modes (HOM), are presented. From the results of the field measurements the coupling (damping) factor and thus the quality factor of the damped resonator can be derived. Measurements of a pillbox resonator with heavily damped TM110-mode (Q < 20) demonstrate the usefulness of the techniques presented, even in this extreme range.
Evidence for an exotic S=-2, Q=-2 baryon resonance in proton-proton collisions at the CERN SPS
(2004)
Results of resonance searches in the Xi - pi -, Xi - pi +, Xi -bar+ pi -, and Xi -bar+ pi + invariant mass spectra in proton-proton collisions at sqrt[s]=17.2 GeV are presented. Evidence is shown for the existence of a narrow Xi - pi - baryon resonance with mass of 1.862±0.002 GeV/c2 and width below the detector resolution of about 0.018 GeV/c2. The significance is estimated to be above 4.2 sigma . This state is a candidate for the hypothetical exotic Xi --3/2 baryon with S=-2, I=3 / 2, and a quark content of (dsdsu-bar). At the same mass, a peak is observed in the Xi - pi + spectrum which is a candidate for the Xi 03/2 member of this isospin quartet with a quark content of (dsusd-bar). The corresponding antibaryon spectra also show enhancements at the same invariant mass.
The results from the STAR Collaboration on directed flow (v1), elliptic flow (v2), and the fourth harmonic (v4) in the anisotropic azimuthal distribution of particles from Au+Au collisions at sqrt[sNN]=200GeV are summarized and compared with results from other experiments and theoretical models. Results for identified particles are presented and fit with a blast-wave model. Different anisotropic flow analysis methods are compared and nonflow effects are extracted from the data. For v2, scaling with the number of constituent quarks and parton coalescence are discussed. For v4, scaling with v22 and quark coalescence are discussed.
Midrapidity open charm spectra from direct reconstruction of D0(D0-bar)-->K± pi ± in d+Au collisions and indirect electron-positron measurements via charm semileptonic decays in p+p and d+Au collisions at sqrt[sNN]=200 GeV are reported. The D0(D0-bar) spectrum covers a transverse momentum (pT) range of 0.1<pT<3 GeV/c, whereas the electron spectra cover a range of 1<pT<4 GeV/c. The electron spectra show approximate binary collision scaling between p+p and d+Au collisions. From these two independent analyses, the differential cross section per nucleon-nucleon binary interaction at midrapidity for open charm production from d+Au collisions at BNL RHIC is d sigma NNcc-bar/dy=0.30±0.04(stat)±0.09(syst) mb. The results are compared to theoretical calculations. Implications for charmonium results in A+A collisions are discussed.
We present the first large-acceptance measurement of event-wise mean transverse momentum <pt> fluctuations for Au-Au collisions at nucleon-nucleon center-of-momentum collision energy sqrt[sNN] = 130 GeV. The observed nonstatistical <pt> fluctuations substantially exceed in magnitude fluctuations expected from the finite number of particles produced in a typical collision. The r.m.s. fractional width excess of the event-wise <pt> distribution is 13.7±0.1(stat) ±1.3(syst)% relative to a statistical reference, for the 15% most-central collisions and for charged hadrons within pseudorapidity range | eta |<1,2 pi azimuth, and 0.15 <= pt <= 2 GeV/c. The width excess varies smoothly but nonmonotonically with collision centrality and does not display rapid changes with centrality which might indicate the presence of critical fluctuations. The reported <pt> fluctuation excess is qualitatively larger than those observed at lower energies and differs markedly from theoretical expectations. Contributions to <pt> fluctuations from semihard parton scattering in the initial state and dissipation in the bulk colored medium are discussed.
The short-lived K(892)* resonance provides an efficient tool to probe properties of the hot and dense medium produced in relativistic heavy-ion collisions. We report measurements of K* in sqrt[sNN]=200GeV Au+Au and p+p collisions reconstructed via its hadronic decay channels K(892)*0-->K pi and K(892)*±-->K0S pi ± using the STAR detector at the Relativistic Heavy Ion Collider at Brookhaven National Laboratory. The K*0 mass has been studied as a function of pT in minimum bias p+p and central Au+Au collisions. The K*pT spectra for minimum bias p+p interactions and for Au+Au collisions in different centralities are presented. The K*/K yield ratios for all centralities in Au+Au collisions are found to be significantly lower than the ratio in minimum bias p+p collisions, indicating the importance of hadronic interactions between chemical and kinetic freeze-outs. A significant nonzero K*0 elliptic flow (v2) is observed in Au+Au collisions and is compared to the K0S and Lambda v2. The nuclear modification factor of K* at intermediate pT is similar to that of K0S but different from Lambda . This establishes a baryon-meson effect over a mass effect in the particle production at intermediate pT (2<pT <= 4GeV/c).
We present a systematic analysis of two-pion interferometry in Au+Au collisions at sqrt[sNN]=200GeV using the STAR detector at Relativistic Heavy Ion Collider. We extract the Hanbury-Brown and Twiss radii and study their multiplicity, transverse momentum, and azimuthal angle dependence. The Gaussianness of the correlation function is studied. Estimates of the geometrical and dynamical structure of the freeze-out source are extracted by fits with blast-wave parametrizations. The expansion of the source and its relation with the initial energy density distribution is studied.
Correlations in the hadron distributions produced in relativistic Au+Au collisions are studied in the discrete wavelet expansion method. The analysis is performed in the space of pseudorapidity (| eta | <= 1) and azimuth(full 2 pi ) in bins of transverse momentum (pt) from 0.14 <= pt <= 2.1GeV/c. In peripheral Au+Au collisions a correlation structure ascribed to minijet fragmentation is observed. It evolves with collision centrality and pt in a way not seen before, which suggests strong dissipation of minijet fragmentation in the longitudinally expanding medium.
Results on high transverse momentum charged particle emission with respect to the reaction plane are presented for Au+Au collisions at sqrt[sNN]=200 GeV. Two- and four-particle correlations results are presented as well as a comparison of azimuthal correlations in Au+Au collisions to those in p+p at the same energy. The elliptic anisotropy v2 is found to reach its maximum at pt~3 GeV/c, then decrease slowly and remain significant up to pt ~ 7-10 GeV/c. Stronger suppression is found in the back-to-back high-pt particle correlations for particles emitted out of plane compared to those emitted in plane. The centrality dependence of v2 at intermediate pt is compared to simple models based on jet quenching.
Azimuthally sensitive Hanbury Brown-Twiss interferometry in Au+Au collisions at sqrt[sNN]=200 GeV
(2004)
We present the results of a systematic study of the shape of the pion distribution in coordinate space at freeze-out in Au+Au collisions at BNL RHIC using two-pion Hanbury Brown-Twiss (HBT) interferometry. Oscillations of the extracted HBT radii versus emission angle indicate sources elongated perpendicular to the reaction plane. The results indicate that the pressure and expansion time of the collision system are not sufficient to completely quench its initial shape.
The pseudorapidity asymmetry and centrality dependence of charged hadron spectra in d+Au collisions at sqrt[sNN ]=200 GeV are presented. The charged particle density at midrapidity, its pseudorapidity asymmetry, and centrality dependence are reasonably reproduced by a multiphase transport model, by HIJING, and by the latest calculations in a saturation model. Ratios of transverse momentum spectra between backward and forward pseudorapidity are above unity for pT below 5 GeV/c . The ratio of central to peripheral spectra in d+Au collisions shows enhancement at 2< pT <6 GeV/c , with a larger effect at backward rapidity than forward rapidity. Our measurements are in qualitative agreement with gluon saturation and in contrast to calculations based on incoherent multiple partonic scatterings.
Transverse energy ( ET ) distributions have been measured for Au+Au collisions at sqrt[sNN ]=200 GeV by the STAR Collaboration at RHIC. ET is constructed from its hadronic and electromagnetic components, which have been measured separately. ET production for the most central collisions is well described by several theoretical models whose common feature is large energy density achieved early in the fireball evolution. The magnitude and centrality dependence of ET per charged particle agrees well with measurements at lower collision energy, indicating that the growth in ET for larger collision energy results from the growth in particle production. The electromagnetic fraction of the total ET is consistent with a final state dominated by mesons and independent of centrality.
We report inclusive photon measurements about midrapidity ( |y| <0.5 ) from 197 Au + 197 Au collisions at sqrt[sNN ]=130 GeV at RHIC. Photon pair conversions were reconstructed from electron and positron tracks measured with the Time Projection Chamber (TPC) of the STAR experiment. With this method, an energy resolution of Delta E/E ~ 2% at 0.5 GeV has been achieved. Reconstructed photons have also been used to measure the transverse momentum ( pt ) spectra of pi 0 mesons about midrapidity ( |y| <1 ) via the pi 0 --> gamma gamma decay channel. The fractional contribution of the pi 0 --> gamma gamma decay to the inclusive photon spectrum decreases by 20%±5% between pt =1.65 GeV/c and pt =2.4 GeV/c in the most central events, indicating that relative to pi 0 --> gamma gamma decay the contribution of other photon sources is substantially increasing.
We present STAR measurements of charged hadron production as a function of centrality in Au+Au collisions at sqrt[sNN ]=130 GeV . The measurements cover a phase space region of 0.2< pT <6.0 GeV/c in transverse momentum and -1< eta <1 in pseudorapidity. Inclusive transverse momentum distributions of charged hadrons in the pseudorapidity region 0.5< | eta | <1 are reported and compared to our previously published results for | eta | <0.5 . No significant difference is seen for inclusive pT distributions of charged hadrons in these two pseudorapidity bins. We measured dN/d eta distributions and truncated mean pT in a region of pT > pcutT , and studied the results in the framework of participant and binary scaling. No clear evidence is observed for participant scaling of charged hadron yield in the measured pT region. The relative importance of hard scattering processes is investigated through binary scaling fraction of particle production.
We report on the rapidity and centrality dependence of proton and antiproton transverse mass distributions from 197Au + 197Au collisions at sqrt[sNN ]=130 GeV as measured by the STAR experiment at the Relativistic Heavy Ion Collider (RHIC). Our results are from the rapidity and transverse momentum range of |y| <0.5 and 0.35< pt <1.00 GeV/c . For both protons and antiprotons, transverse mass distributions become more convex from peripheral to central collisions demonstrating characteristics of collective expansion. The measured rapidity distributions and the mean transverse momenta versus rapidity are flat within |y| <0.5 . Comparisons of our data with results from model calculations indicate that in order to obtain a consistent picture of the proton (antiproton) yields and transverse mass distributions the possibility of prehadronic collective expansion may have to be taken into account.