nucleon

Nuclear Physics @ North-Holland

A433 (1985) 7 13-742 Publishing Company

PARTIAL-TRANSPARENCY EFFECTS IN HEAVY-ION COLLISIONS AT ENERGIES OF THE ORDER OF 1 GeV/NUCLEON W.B.

IVANOV

7Ite I. V. Kurchatov Institute of Atomic Energy, Moscow, USSR I.N. MISHIJSTIN* The Niels Bohr Institute, University of Copenhagen, DK-2100 Copenhagen 0, Denmark and L.M. SATAROV 7he I. V. Kurchatov Institute of Atomic Energy, Moscow, USSR Received 5 October 1983 (Revised 8 June 1984) taking into account the Al&tact: A model of heavy-ion collisions at energies Eub- 1GeV/nucIeon effects of incomplete statistical equilibrium in highly-excited nuclear matter is presented. The collision process is considered as the interaction of two flows of nucleons decelerating each other. This process is described within the framework of a relativistic kinetic approach employing the Fokker-Planck approximation. Assuming two-flow nonequilibrium the momentum distribution function is represented as a sum of two maxwellian distributions displaced by the average relative velocity of flows. Equations for time evolution of space-averaged velocities and internal energies of flows are derived. These equations contain a single model parameter, i.e. the effective deceleration length A,,. Using the firestreak model geometry inclusive cross sections of protons and composite particles (d, t) in the reactions Ne + NaF, Ar + KCI and Ar+ Pb are calculated at various values of E&,. In contrast to the tirestreak model, a complete stopping of colliding tubes in the centre-of-mass frame was not assumed in the present calculation. Composite particle spectra are calculated on the basis of the coalescence model. The theory is thoroughly compared with the experimental data. The approach suggested allows us to reproduce a two-humped structure in the rapidity distributions of the secondary particles. The experimental data analysis leads to the value A, = 8 fm for E,,, = 0.8 GeV/nucleon, which is in good agreement with the estimation based on experimental NN cross sections.

1. Introduction The study of hadronic-matter properties in experiments with heavy ions at high energies (- 1 GeV/nucleon) is of great interest from both the theoretical and experimental points of view. Sufficient experimental data on the interaction of heavy ions of medium mass (up to Fe) with various targets [see refs. ‘*‘)I have already been obtained, and in the near future data on high-energy beams of heavier ions (up to U) will become available. The interest in collisions of the heavier ions is due to the obvious fact that in this case macroscopic properties of hadronic matter, i.e. the

*

Permanent

address:

The I.V. Kurchatov

Institute 713

of Atomic

Energy,

Moscow,

USSR.

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Yu.B. Ivanov et al. / Partial-transparency

peculiarities of the equation of state, as well as hydrodynamic-type collective motions, are expected to be most strongly pronounced. This requires a large enough number of particles so that surface effects as well as phase-space constraints do not play an important role. Furthermore, the assumption that thermodynamic and chemical equilibrium is established in the process of collision acquires sufficient grounds only in this case. Proceeding from the same considerations as in the neutron moderation theory 3), one can evaluate an average number of collisions r+ required by each nucleon to thermalize the initial momentum “). Assuming that the thermalized source moves with the centre-of-mass velocity of the overlapping parts of nuclei, one can easily estimate VT= 2-4, at the bombarding energy Elab - 1 GeV/nucleon. At this energy a nucleon mean free path in nuclear matter of normal density is about 2.5 fm [ref. “)I; therefore a slab of nuclear matter 5-10 fm thick is needed to quench the initial momentum of colliding nuclei. This value is comparable with radii of medium and heavy nuclei. Relatively light nuclei, such as C, Ne and Ar, are not “thick” enough to stop nucleons in c.m. systems and partial-transparency effects should be expected in the case of their collision. Recently an interesting kinematic analysis ‘) of experimental data “) on inclusive cross sections of protons and composite fragments in nuclear reactions at high energies was performed. It was found that rapidity distributions of reaction products with a low transverse momentum have a specific two-humped structure. It manifested itself most clearly in the case of composite particles. We believe that this structure is attributable to the nuclear partial transparency hindering the establishment of a local thermodynamic equilibrium. The aim of the present paper is to investigate, in the framework of the kinetic theory, the evolution of the system of colliding nuclei from the initial nonequilibrium state to that with partial or total thermodynamic equilibrium. Special attention is paid to the description of peculiarities in secondary-particle spectra, which arise due to the partial transparency of colliding nuclei. It is clear from the following that by studying these peculiarities as a function of the mass number of the colliding nuclei, the beam energy and the reaction impact parameter one can obtain information about the kinetic properties of hadronic matter. To show clearly the place of our approach amongst the many others that have dealt with nonequilibrium features of high-energy heavy-ion collisions, we think a brief historical review will be appropriate here. The simplest description of highenergy heavy-ion collisions is presented by the nuclear fireball model 7*8),assuming that the overlapping parts of colliding nuclei form a thermodynamically equilibrated object, the fireball, which emits secondary particles. Further theoretical and experimental investigations gave ground to the idea of incomplete thermalization of the initial relative momentum in the case of higher bombarding energies and lighter colliding nuclei. It was first observed in ref. ‘) that the description of experimental data on Ne + U collisions at 2.1 GeV/nucleon becomes better if two fireballs were

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introduced instead of one and thermalization of only 25% of initial relative momentum was assumed. Later, in ref. 9), the simple version of the two-fireball model was successfully applied for the calculation of proton energy distributions in C+C collisions at 800 MeV/nucleon. In refs. **lo)the stopping power of nuclear matter was estimated using the known free nucleon-nucleon cross section. The two-fireball model was justified and essentially developed. By detailed calculations of secondaryparticle spectra it was shown that partial transparency of nuclei can explain the noticeable angular anisotropy of inclusive cross sections in the reactions C + C and Ne + NaF. Similar results were obtained also in the presently very popular firestreak model ‘I,‘*) assuming local thermodynamical equilibrium but taking into account the spatial nonuniformity of temperature and average velocity fields. In ref. 13)this model was generalized for the case of partial transparency of nuclei in relativistic collisions. Partial stopping of juxtaposed tubes was considered in a way very similar to our approach. A more sophisticated approach containing both the space-time development of the system and nonequilibrium effects is the two-fluid hydrodynamical model suggested in ref. 14). In this approach the collision of heavy ions is considered as the interaction of two fluids containing projectile and target nucleons. In contrast to conventional hydrodynamics, assuming instantaneous establishment of thermodynamic equilibrium, a friction between the fluids (nucleon flows) is introduced that results in their mutual deceleration accompanied by gradual heating and, finally, their stopping. Unfortunately, detailed caclulations were performed only for the Ne + U system at 400 MeV/nucleon. In this case the difference from conventional hydrodynamics turned out to be not so marked. It will be shown below that conceptually our approach is close both to the two-fireball model and to the two-fluid hydrodynamical model but contains a number of important distinctions. A number of microscopic approaches have been developed, in which the process of nucleus-nucleus collision is divided into a series of nucleon-nucleon collisions. These models are evidently suitable for studying the time evolution of the particle momentum distribution. Many existing realizations of this approach differ in intranuclear-cascade scenarios as well as in the method of describing the nucleon-nucleon interaction. In the first cascade models this interaction was imitated by means of the cross sections 15-17).Later, potential cascade models with a more or less realistic nucleon-nucleon potential 18,‘9) as well as the hybrid model *‘) were developed. All these models have so far been formulated on the classical level; sometimes a simulation of Pauli’s principle is introduced. Despite their differences, all the microscopic models come to the conclusion that during the whole compression stage a considerable nonequilibrium of the nucleon momentum distribution takes place, which manifests itself in a noticeable difference between the mean values of the longitudinal and transversal momenta 17,18).A thermodynamic equilibrium is established, if at all, only in the case of sufficiently heavy ions and at the stage immediately prior to the decay of the hot zone.

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Of great relevance to our present work are studies based on relativistic kinetic equations. In ref. ‘I), as well as in the original rows-on-rows model 22) and other related works, transport-like equations for spectral distributions were derived and the equilibration process in relativistic heavy-ion collisions was investigated in great detail. The first application of the Fokker-Planck equation for calculating the energy spectra of secondary-particles interacting with a thermal bath was made in ref. 23). In ref. 24) a Fokker-Planck equation for the partial stopping of the tubes was derived and its analytical solution was found in a two-maxwellian parameterization of the distribution function for a spatially uniform system. Finally, it should be noticed that the effects of partial transparency became known long ago in hadron-hadron and hadron-nuclear collisions at high energies [see e.g. ref. “)I. They appear as the availability of so-called leading particles which take a significant part (-f) of the initial longitudinal momentum. The relativistic kinetic-equation approach is developed below to study noncomplete thermalization effects in high-energy heavy-ion collisions. The formulation of the problem and an analysis of the approximations used are given in the next section. Equations describing the slowing-down and the heating of two nuclear matter slabs in the Fokker-Planck approximation are derived in sect. 3. Sect. 4 considers the procedure employed for calculating inclusive spectra of secondary particles. In particular, two formulations of the coalescence model used for calculating composite-particle spectra are presented. In sect. 5 the results obtained are discussed and analyzed. Sect. 6 contains conclusions and proposals for future experiments. 2. The model The aim of this paper is to calculate the inclusive spectra of protons and complex particles (d, t, 3He) within a broad interval of particle energies and emission angles. Formation of nuclear fragments is treated on the basis of the coalescence model (see below). In our semiphenomenological approach we explicitly take into account the possibility of incomplete thermalization of the initial momenta of colliding nuclei. In fact, this approach is close to the two-fluid hydrodynamical model 14)but is simpler in its calculational procedure. As in the firestreak model, we assume that only the overlapping parts of nuclei suffer considerable interaction in the process of nuclear collision. Moreover, this interaction is subdivided into interactions of separate tubes (streaks) arranged opposite to each other. The transverse interaction between the tubes is disregarded. Unlike the firestreak model, we do not assume a complete stopping of the colliding tubes in their centre-of-mass frame. It is evident that the degree of deceleration of colliding tubes is determined both by their length and the efficiency of friction due to a cascade of nucleon-nucleon collisions. As has already been noted, in the range of initial energies considered the stopping length at which the transition from the nucleon-directed motion to a chaotic (thermal)

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one takes place should amount to 5- 10 fm, which is comparable to the size of medium and heavy nuclei. Therefore, we assume that as a result of incomplete stopping of tubes, pa~icularly those of a short length, the spatial dist~bution of nuclear matter looks like two bunches of nucleons. The velocities and temperatures of these bunches should be found from dynamical considerations and conservation laws. Somewhat similar concepts were employed in ref. 13). Our purpose here is to give the simplest formulation of the problem, which does not require cumbersome numerical calculations. In the next section, on the basis of kinetic theory, we will show that the mutual deceleration of tubes can be described by the equation of time evolution of their average relative velocity VJt):

J,4_ d Vre, m=-_

Zp+Zt dt

t

1 Ad

v2

m’

np(5th(s t)

dz

n;

Here and below the indices p and t correspond to the projectile and target nuclei, respectively, ZP,tare colliding tubes lengths normalized to the normal nuclear density n, = 0.15 fmF3, np,t are density distributions in the tubes (in their rest frames of reference), z is a coordinate along the tube axes, and Ad is the effective deceleration length depending, in general, on V,,. The factor Z,ZJ(Zr-64) stands for the reduced mass in terms of length units. The r.h.s. of eq. (1) defining the deceleration efficiency, as should have been expected, is proportional to the tube-density overlapping integral. Within the kinetic approach the expression for the effective deceleration length can be found through hd=2/nOuti,

(21

where a,, is the transport cross section of nucleon-nucleon interaction in the medium. The use of vacuum nucleon-nucleon cross sections at Z&, = 800 MeV results in the estimate Ad = 8 fm. However, it should be stressed that due to the medium polarization effects the effective scattering cross section can differ considerably from the vacuum one. Therefore, A,.,should be treated as a model parameter whose value is to be found by comparing calculations and experimental data. To avoid any additional model parameters, let us assume Ad (V,,,) = const. As shown below, in this case the following relation between the initial (0) and final (fin) relative rapidities of interacting tubes can be derived: tanh t&v, - y&d = tanh t&y, -Y&O) exP ( - (1, + &)/Ad) .

(3)

From this equation it is clear that the final rapidities become practically equal, (Y, - Y*h, = 0, when the tube’s sum length ZP+Z,is greater than or of the order of the deceleration length Ad. To describe the tube state after the interaction, one should know how the dissipated energy of the relative motion, i.e. the intrinsic excitation energy, is distributed between

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the tubes. From eq. (3) and conservation of the total 4-momentum,

(P;+Pf),,=(P;+P:),,

i=O, 1,2,3,

(4)

the rapidities y,, and yt and internal energies (per particle) ep and Edof each tube can be determined. After these values have been determined one can introduce single-particle distribution functions and calculate nucleon spectra in the framework of the firestreak-model geometry (see sect. 4). The kinetic treatment results in (here and below h=c=l)

where the upper and lower signs refer to the p- and t-tubes, respectively, A( YO,Ys,) =fi< Y,Q,“-l)“‘[arctan

(f( YO- 1))1’2-arctan

(i( Y,,- 1))“‘2],

(6)

mN is the nucleon mass, and YO,fi,= cash (y, - J+)~,~~.The energy-conservation law, eq. (4), has been taken into account in eq. (5). In the case of high transparency, i.e. when after the interaction the tube velocities change negligibly, ( YO- Y,,)/ YO% 1, the excitation energy per particle turns out to be inversely proportional to the tube lengths: l,( E+,- mN) = Zt( E,- mN).This situation corresponds to the two-fluid hydrodynamics as was formulated in ref. 14) when, as a result of flow friction, only the “generation” of flow internal energies takes place but heat exchange between the flows is completely absent. As shown in the next section, the derivation of the two-fluid hydrodynamical equations based on the kinetic theory results in an additional, as compared with ref. 14),term corresponding to the heat exchange between flows. It tends to equalize the flow temperatures so that in the case of a complete stopping ( Y& = 1) the internal energies per particle of two flows become equal to each other and to the corresponding value in one-fluid hydrodynamics. In the firestreak picture, during the deceleration process the shorter tube’s internal energy first grows to a maximum, falls and reaches the longer tube’s excitation energy per particle. Expressions (3), (5) and (6) completely determine the tube states after their interaction. A microscopic description of composite-particle production mechanisms in heavy-ion collisions is beyond the scope of the present paper. Therefore, compositeparticle yields were calculated using a simple semiphenomenological version of the coalescence model 26727),which agrees with the experimental data fairly well 6). The explicit consideration of composite-particle formation is important because a considerable number of protons leave the reaction volume in nuclear fragments, and in calculating the observable proton spectra one should subtract the contribution of these bound protons. Moreover, as shown below, transparency effects manifest themselves most pronouncedly in the nuclear fragment spectra.

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At bombarding energies of - 1 GeV/nucleon, inelastic channels of NN interaction are already open. Mainly, these are the A-isobar and rr-meson production. Though the number of A-isobars and w-mesons produced is ~mparatively small 4), they cause a noticeable decrease of the tube temperature. In our model the inelastic processes are taken into account implicitly by inclusion of the A-isobar and r-meson contributions to the equation of state of hot nuclear matter. The concentrations of A-isobars and Ir-mesons are calculated, from the condition of thermodynamical equilib~um, in each tube separately (see sect. 4). One of the most essential approximations used below is a replacement of inhomogeneous distributions of local velocity U and temperature T by the values averaged over the tube volume. Its accuracy is difficult to estimate. In reality, due to the finite velocity of a perturbation more and more remote from the contact point, layers of matter will successively be involved in the interaction. As a result, the Uand T-profiles will be strongly inhomogeneous along the tube axis. At the expansion stage, when the overlapping region starts to contract, the degree of inhomogeneity of velocity, temperature and pressure distributions decreases. In each tube the pressure gradients, viscosity and thermal conductivity tend to make the U- and T-distributions more homogeneous. Actually, we are interested only in the asymptotic velocity dist~bution which is formed at the later stages of expansion when the velocity and temperature variations in a separate tube are certainly lower than the velocity and temperature spread of different tubes. Transition to mean values at the tube interaction stage should be understood only as a way of calculating a mean value of the dissipated momentum and energy. In ref. *), in calculating the stopping power of nuclei, the tubes were divided into separate narrow blocks subjected to successive pair-wise interactions. This approach can hardly be accepted as a sufficient one, since in this case the inverse distribution of block velocities takes place (the backward blocks move faster than the forward ones) and the subsequent “mixing” of matter should be taken into account. The disadvantage of our model, as well as of the firestreak model, is the disregard of transversal degrees of freedom, namely, friction between the adjacent tubes has not been taken into account, as well as the transversal expansion of tubes. The “spectator” (nonoverlapping parts of nuclei) excitation and their collective motion under the action of hot-participant pressure are also ignored. Apparently, all these effects will be less important under the conditions of partial transparency of nuclei, than under the conditions of the~odynamical equilib~um. Only a compa~son with experimental data may tell us how important transversal collective motion effects are. It is clear that most of the approximations used can be avoided by solving the two-fluid hydrodynamical equations numerically. However, up till now this approach has not been so popular due to the great computational difficulties. As in the firestreak model we do not explicitly consider the contribution of direct processes due to single collisions of nucleon pairs from the projectile and target nuclei. It is known that this contribution is enhanced in the case of light nuclei and

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peripheral collisions ‘*l’). On the other hand, the two-humped structure of the secondary-particle rapidity distributions (most pronounced in nuclear fragment spectra) is also observed in collisions of sufficiently heavy nuclei like Ar, in which direct processess give a relatively small contribution 2’). Moreover, the direct knockout of composite particles and their formation in pick-up-like process involving direct nucleons are highly improbable events at the bombarding energies considered 6*28). 3. Flow friction equation In this section the derivation of the nucleon-flow friction equation will be presented. The present consideration is more an illustration of the physical concepts comprising the model, than a strict grounding of our approach. We proceed from the assumption that the collision process of two-nucleon flows is described by the Boltzmann relativistic kinetic equation 29,30)t

where m is a nucleon mass (the index N is dropped in this section), du/dt(s, t) is the invariant differential cross section of nucleon-nucleon scattering in nuclear matter, which, generally speaking, may differ from the respective value in vacuum, s = ( p +P,)~ is a squared sum of the colliding nucleon 4-momenta, while t = (p - p’)’ is a square of the momentum transfer. The distribution function (DF) of nucleons f(p, x) is essentially a relativistic invariant 3’) being the particle density in a unit volume of the phase space. Eq. (7) was used earlier in ref. 32) for calculation of the kinetic coefficients of hot nuclear matter. Let us consider the characteristic features of free nucleon-nucleon cross section in the lab energy range Eta,,- 1 GeV. At Elab = 800 MeV the following fit formula 33) is used for the experimental elastic pp cross section in the c.m. scattering angle range 0 s 8c.m.S 4~: da/dt

= A exp ( - Bt + Ct’) ,

(8)

where t=-mmE,,,,(l-cos 19,,.), A=126.20*9.6mb/(GeV/c)2, B=-6.51k0.34 (GeV/c)-2, and C = 3.56 *to.39 (GeV/c)-4. Hence, we obtain da/dt = 126 mb at 0c.m.= 0, and da/dt = 7 mb at 8,.,. = $r, Thus, at Elab- 1 GeV the pp scattering is strongly forward-backward peaked. In this case it is natural to use the Fokker-Planck expansion in the collision integral, r.h.s. of eq. (7), assuming that the transferred momentum is small compared to the total momentum. ’ Below we use the following

form of the metric tensor g,, (i, k = 0, 1,2,3):

g,, = 1, g,,

= g,, = g,, = -1.

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Further on, we regard the colliding particles as identical, without distinguishing neutrons and protons. By definition, the particles scattered to the forward hemisphere (19,.,. < &r) are attributed to the projectile nucleon flow, and those scattered to the backward hemisphere (e_.>$r) to the target one. Below, these flows will be denoted by the indices p and t, respectively. So each flow is formed from both the initial particles and those subjected to deviations to large angles and “transferred” from one flow to another. In classical mechanics the measured scattering cross section of identical particles is equal to a sum of scattering cross sections in two diametrically opposite directions (in the c.m. frame):

, $+~.)={~w.~~} +{-$+-km.)) proj p-i where {dC/dR}p,oj is the cross section of a projectile particle scattering. Transforming the collision integral so that it contains only the observable cross section, one can see that only forward-hemisphere angles (8,,. s $r) are taken into consideration. This corresponds to -f( s - 4m2) G t < 0. The latter conclusion remains true in quantum mechanics, too. Moreover, in the case of quantum mechanics, due to interference effects, it is impossible to express the observable cross section in the form of a sum of projectile and target particles cross sections. Therefore, the region of scattering angles e,.,. should be restricted to the forward hemisphere from the very beginning, if the observable scattering cross section of identical particles is used in the collision integral. In this approach the mean longitudinal momentum transfer per nucleonnucleon collision is

The larger the angular anisotropy of the nucleon-nucleon cross section is, the less this value becomes. In fact, the Fokker-Planck expansion is performed in this parameter squared. To carry out the Fokker-Planck expansion in an explicitly covariant form, it is convenient to introduce the DF (e(z) = f( 1 + signz)) F(p, x) =NP,

x)W-

mz)e(pO) ,

for which the kinetic equation is written as follows:

where

-&i(P’F)= - I W(PPIIP’PII)M-, -f’fi)

d4p,d4p’d4p;,

(9)

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and the arguments p and p” of the functions f are no longer constrained by the condition (p”)’ -p2 = m2 but are considered as independent variables. Now, in eq. (10) all 4-momentum components are equally comprised and we can perform the expansion in the 4-momentum transfer q =p’-p up to terms of the order of q2 in the same way as was done by Landau 34) in the nonrelativistic approach. In this case we obtain

(11)

Bik(p,PJ ,

I’(P) = BIk(p,p,)=2

(12)

qiqk-!~[(p.p,)2-m4]S(q2+2q*P)S(q2-2q*p1)d4q. I

(13) From eq. (13) it is clear that piBik -P,$?‘~ - q2qk are small values of order greater than q2. With this in mind we arrived at eq. (12). The S-functions 6(p2- m’) and 6(pfm2) were separated from W (ppl jp’p:) and brought under the derivative signs af/apkand af,/ap", since Bikas( p2 - m2)/apka Bikpk= 0. Proceeding from the expression (13) one can easily find an explicit form of the tensor Bik. First, a symmetrical tensor wik should be constructed from the 4-vectors pi, pi and tensor gik, which fulfils the conditions piwik = Pl,iWik = 0, and then, having calculated the scalar gikBik, the normalization of the tensor should be defined. AS a result, we obtain Bik(p, pJ = Wik

wikB(s),

(14) Pl)(P’p:+PfPk)

=-[(P*P1)2-P2P:lgik-(P:P$k+P2PfP:)+(P

c%(r) ,

=27T

-$!&

( 1- cos K.,.) sin %,. de,.,. .

9

(15)

(16)

It should be emphasized once more that the transport cross section in eq. (16) is determined by the integration over the hemisphere 13~.~.~&r only, due to the fact that the particles are identical. Now let us consider the collision of two collinear nucleon flows. The initial state of the system (before collision) is represented by two equilibrium distributions separated in space. We assume that in the course of collision the two-flow type of the initial nonequilibrium survives, i.e. the DF of the system is represented by the

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sum of two displaced Maxwell distributions 3s) [a very similar parametrization used in ref. ‘“)I:

f(P, xl .L(P, x) =

n,(x) 4rmZT,K2(m/

=_&(p, x)+ft(p,

T,)

ew {-

x) ,

Uh(x)Pil

T,(X)),

was

(17) (Y=p,t.

(18)

In eq. (18) n,(x) is the local particle number density of the a-flow in the local rest frame; T,(x) and V:(x), the temperature and the hydrodynamical 4-velocity of the a-flow; K,(z), the McDonald function. By representing the DF in the form of eqs. (17) and (18), we assume the in-flow equilibration process to be much faster than the relaxation of the two-flow nonequilibrium. There are some arguments for such an assumption. First, the slow relaxation of the two-flow nonequilibrium (i.e. the flow slowing down) is mainly due to the considerable angular anisotropy of the nucleon-nucleon cross section (see eq. (8)). This anisotropy results in a considerable characteristic length of longitudinal momentum dissipation (- 10 fm at Elab- 1 GeV/nucleon). Due to the same reason only small nonequilibrium perturbations are created inside an individual flow. Pauli blocking leads to an additional suppression of these perturbations. Moreover, their relaxation is fast, as a result of in-flow particle collisions. The high efficiency of in-flow collisions is a consequence of a large absolute value and the isotropy of the NN cross section at low relative energies. The three-flow distribution seems to be an even more realistic approximation 36). In this approach the p- and t-distributions correspond to two groups of leading particles, whereas the central distribution (c) represents a group of inelasticallyscattered particles. In this picture the slowing-down of the p- and t-flows, as well as their attenuation with the rise of the c-distribution, takes place. It is worthwhile to mention that the large-angle scattering can be enhanced due to the interaction of particles with collective modes and dynamical instabilities 37). Actually, the inelastic nucleon collisions with large kinetic energy losses due to the production of new particles (A-isobars and r-mesons) will be the main source of the cdistribution. At bombarding energies Elab - 1 GeV/nucleon the equilibrium concentrations of A-particles and pions are still relatively low 38,39).In the first approximation these inelastic processes can be taken into account by attributing corresponding large energy and momentum transfers to the entire p- and t-flows. It will increase the stopping power of colliding tubes, i.e. decrease the effective stopping length. The presence of A-particles and pions in each tube can be accounted for by the inclusion of corresponding contributions to the hot nuclear matter equation of state (see below). We hope that in this way even the two-flow approximation will give qualitatively correct results. Using the two-flow representation of the DF, eqs. (17) and (18), one can obtain the set of kinetic equations for fp and ft. As a consequence of the maxwellian form

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of the p- and t-distributions, only the cross terms corresponding p- and t-particles remain in the collision integral: a(p’F,)/axi

= -al;/@‘)

to collisions of the

a=p,t,

(19)

Z;(P)= d4zOpa&,/apt-F,,, aFplapkWk(p,p,). (Zf( p) is defined similarly.) Let us introduce the particle currents and energy-momentum

(20)

tensors:

J;(x) =

F,(p,

x)p’ d4p = n,(x)Uh(x) ,

(21)

T:(x) =

F,(p,

x)p’pkd4p,

(22)

where (Y= p, t. From eqs. (19) and (20) we obtain

(23)

aJ;/ax’=O,

aT;/aXk= -

FpF,,l(pi-p;)sB(s) d4p d4p, .

(24)

The set of equations (23) and (24) will be significantly simplified if one takes into account that sB(s) is a much smoother function of p and p1 than F and F,, and removes it from the integral at the intermediate point so= mz (Up+ Ut)*, where F,,( p, x)F,(p,, x) has a maximum. Then, the set of equations (23) and (24) is reduced to the two-fluid hydrodynamics equations. In the zero-order approximation in T,/m < 1 the r.h.s. of eq. (24) coincides with the flow interaction term of ref. i4): soNso)

I

&F,,I(P’

-P:)

d4pd4p,= %W;W

D(x) = sowo)n,(xh(x)~

- UfWI, (25)

The use of the Fokker-Planck expansion has made it possible to derive the flow friction coefficient D(x) in an analytical form. Later on, we treat the r.h.s. of eq. (24) more accurately, i.e. up to the first order in Ta/ m. This is necessary for a proper account of the heat exchange between the flows, which leads to the interflow thermal equilibration when the flows decelerate significantly. We roughen our approximation even further, as compared with the hydrodynamical one, by considering the time evolution of macroscopic quantities averaged over some space region. Let us consider a one-dimensional problem of collision of two (p and t) tubes of nuclear matter. At a fixed time we describe this system by the total number of particles in each tube N,, the space-averaged 4-velocity LJb and the internal energy

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per particle a, of each tube: N, = E,U;N,

=

I

I

Jo,(x) d3x,

(26)

p,‘(x) d3x,

(Y=p,t.

(27)

Performing spatial averaging in eqs. (23) and (24) we obtain the equations for the averaged quantities N,, U, and E,: dN,ldt

dN,/dt = 0,

=O,

(28)

N,dP~/dt=-D(~)(P:,-P~)/(E~E~), N,dPf/dt

(29)

=-D(t)(l’f+‘;)/(~~~~),

(30)

where PL = E,U& (Y= p, t, D(t) = j D(x) d3x. When deriving eqs. (29) and (30) the local functions E,(X) and U:(x) in the r.h.s. were replaced by their averaged values according to eqs. (26) and (27), and the smallness of the parameter (a, - m)/m =S1 was used. This condition is equivalent to that of Tu/ m < 1, which is always satisfied in practice. Eqs. (29) and (30) are derived in the first-order approximation in (% - m)/m. The set of equations (28)-(30) allows us to get separate equations for the time evolution of internal energies and the relative velocity of tubes. To obtain equations for E,, one should multiply eqs. (29) and (30) by Pi and Pf, respectively, and take into account the condition PLP,,i = ~2 (CYis fixed). Thus, one obtains

,

(31)

(32) The r.h.s. terms in eqs. (31) and (32), proportional to (Up- UJ’, describe the transition of the relative motion kinetic energy into the internal energy of flows. The terms proportional to ( ap - et) yield the heat exchange between the flows. In order to obtain the equation for the relative velocity one should subtract eq. (30) from eq. (29) and multiply the difference by (Pk - Pi). Neglecting terms proportional to (a, - m)/m < 1, and making use of eqs. (31) and (32), one gets

tm2; (Up-

q2=

-D(t)

(

+++ P

>

[(Up-

t

u,>“-a( up- t-J,)“].

(33)

The transition to eqs. (31)-(33) has the advantage of separating the equations for the relative motion and the heating of the tubes. Let us introduce the flow rapidities yp and y, along the beam direction (z-axis): Ub = (cash y,, 0, 0, sinh y_) ,

cY=p,t.

(34)

726

Yu.B. Ivanov et aL / Partial-transparency

Rewriting the values Ub and D(t) in eq. (33) in terms of yn, we find the equation for the relative velocity Vrel= tanh (y, - y,):

($+j$ j-n,(x)n,(x) d3x,

d Vre,

-=-btrV;e, dt

(35)

which is the flow friction equation in question. Finally, introducing the tube lengths in the tube rest frames IP and Z, (Nu = nob for tubes of a unit cross section), we rewrite eq. (35) in the form

(36) where A,.,= 2/ noa,,. Further on, the parameter A, (the deceleration length) is assumed to be a VT,,-independent value. It was shown above that the estimation of Ad on the basis of the experimental pp cross section at Elab = 800 MeV yields Ad= 8 fm. In this estimation the fit of the experimental elastic pp cross section, eq. (8), was used and the inelastic pp cross section dominated by the process pp+ NA was assumed to be isotropic in the centre-of-mass frame. The numerical value of the flow friction coefficient used in the two-fluid hydrodynamical calculations 14)brings about the same estimation of Ad. In the case of a developed two-flow regime the deformations of density distributions in each flow are small. On the other hand, in the opposite limit, when a complete stopping and fusion of colliding tubes occur, our approach coincides with the firestreak model in which the maximum compression of nuclear matter is not so important. Therefore, a reasonable assumption is that the densities r+,,*do not change significantly during the collision: nP = n, = no. At the Lorentz transformation from the laboratory reference frame to the target-instantaneously-comoving frame the time and space intervals change as follows: di= cash yt dt and dz’= dz/cosh yt. In this new reference frame the flow friction equation looks like that in any inertial frame (z and t are just replaced by i and ?). The overlap integral in the r.h.s. of eq. (36) can be represented in the form

I

np(i Me

4

t3

di=

2

co& (Y, - YJ ’

(37)

where the factor l/cash (y,- y,) takes into account the Lorentz contraction of the projectile in the target rest frame, and Z is the geometrical overlap of noncontracted tubes at time t. For the sake of simplicity, we consider the case when I, < 1,. Then I = I, and eq. (36) takes the form (38) where dz,, = V,, d?. The integration over dz,,, is performed from 0 to Z,.By solving

727

Yu.B. Zvanov et al. / Partial-transparency

eq. (38) we obtain the final relation (3) given in the previous section. Rather bulky calculations result in the same relation in the general case of arbitrary Z, and I,. To find the tube heating in the course of the deceleration, we must use eqs. (31) and (32). In these equations the relative 4-velocity squared (UP- UJ2 = 2[1 -cash (yr-y,)] should be treated as a function of time known from the solution of eq. (33). Therefore, to avoid the tiresome Lorentz-noninvariant integration over t, it is convenient to introduce a new independent variable, Y = cash (y,-yJ, on the basis of eq. (33): 2

D(t)dt=-+$m P

-

t

dY

(39)

Y2-1’

Using eq. (39) the set of equations (31) and (32) can be transformed

,

into (40)

(41) The initial conditions for this set of equations are

where Y0= cash (yr-yYt),,. Eqs. (40) and (41) should be integrated from Y0 up to Yr,, = cash (Y, -~&in. To simplify the calculation, one should use the integral of motion of these equations which is a consequence of the total 4-momentum conservation: (Np~pUp+Nt~tUt)2=(Np~p)2+(Nt~,)Z+2NpNt~petY=const.

(42)

A simple but lengthy calculation, which consistently employs the approximation (E, - m)/m QC1, results in the final expressions for ~r,~,eqs. (5) and (6), given in the previous section. 4. The calculation of secondary particle spectra The inclusive cross sections of secondary particles were calculated using the geometrical concepts of the firestreak model 2*11,12).We introduce the nuclearcollision impact parameter 6 which is a two-dimensional vector within the plane orthogonal to the beam axis z. It is assumed that only nucleons from overlapping parts of colliding nuclei, S(b), take part in the reactioh at fixed b. If s is a two-dimensional vector directed from the centre of the projectile nucleus and lying within the plane orthogonal to the beam axis, then S(b) may be defined as S(b):

s
where R, and R, are the effective radii of the projectile and target nuclei, respectively.

728

Yu.B. Ivanov et al. / Partial-transparency RP

AP

Fig. 1. The nuclear collision geometry within the firestreak model. A and B are the front and side views relative to the beam direction, respectively. The overlap region is divided into pairs of interacting tubes.

The overlapping parts of nuclei are divided into pairs of tubes with cross section uNN = 40 mb arranged opposite to each other (see fig. 1). It is assumed that each pair of tubes interacts and gives a contribution to the momentum distribution independently of the others. In the calculation we take into account the diffuseness of the nucleon density distribution ncr (s, z) at the nuclear surface. Actually, the formulae contain the tube length la(s) =;

n,(s, z) dz.

(43)

In the calculation of I,(s) we used the approximation suggested in ref. 40). To avoid dealing with tubes containing less than one nucleon and, thus, to provide for the proper normalization of the total inclusive cross section a,,, of the charged particles yield, we eliminate the contributions of the far peripheral tubes. In our approach the expression for a,,, is

where 2, and A, are the charge and the mass number of the (Y-nucleus, respectively. The values of the effective radii R, are determined by fitting the calculated o,,, to the experimental one. Since the experimental values of a,,, are known with an accuracy of 20-30% [ref. 6)], the R, values were varied within - 10% to achieve a better fit of the experimental spectra (see below). As was mentioned above, we proceed from the assumption that the equilibrium in each individual tube is maintained during the whole interaction process. At the bombarding energies considered the nucleon-nucleon collisions result in excitation of the m-mesons and lowest baryon resonances. In the first approximation it is

729

Yu.B. Ivanov et al. / Partial-transparency

sufficient to take into consideration the A-isobars with mass mA= 1232 MeV. The nucleons and A’s are assumed to be in thermal equilibrium in each tube characterized by the temperature T, baryon density n or chemical potential /_L. Thermodynamical functions and momentum distributions of particles are defined in terms of the single-particle DF f”‘(p, x) (i = N, A). In the tube rest frame, neglecting the degeneracy effects, fCi) may be written as follows: {dT(YP)}.

f”‘(p, x) =&exp

(45)

Here E”‘(p) = (m2+p2)“2, and mi and gi are the mass and the spin-isospin degeneracy factor of particle i (g, = 4, gA = 16). The chemical potential p is determined by requiring the summary density of N’s and A’s to be equal to the given total baryon density n: J,

(46)

j- d3pf”’ = n -

To first order in T/m, < 1 one can find an explicit expression for /.L[ref. *‘)I: g,(2r)-3

ePLIT= n e”N’=/K(T),

(47)

where K(T)

= (2rmNT)3’2(1

+gT/m,)[l+

cA( T)].

The value 312

ev

{-

(mA

-

mN)/

Tl

is the equilibrium ratio of the A- and N-densities. The internal energy per baryon is defined as

.A

dJp

1

n

i=N,A

E(i)(p)f(i)

(48)

J

Taking account of eqs. (45) and (47) we obtain the equation of state [see ref. “I)]: ~(T)=rn~+;T+(rnA-mN)

CA(T)

l+cA(T)’

The third term in the r.h.s. of eq. (49) is the contribution of the A-production. In actual calculations the contribution of the 7r-mesons has been taken into account according to ref. 4’) (it becomes noticeable at Elab> 2 GeV/nucleon). The temperatures T, of each pair of tubes after their interaction are determined from eq. (49) by means of the internal energies E, = E( T,)(a =p, t), the values of E, being known from eqs. (3) and (5). Here and below we imply that the E, and T, as well as the averaged tube 4-velocity U,, are s-dependent.

730

Yu. B. Ivanov et aL / Partial-transparency

In an arbitrary reference frame the DFf”’ is given by eq. (45) (cf. eq. (18)) when substituting T = T,, p = CL, and replacing EC’) by E&“‘=(p”‘U,)=~cosh(y’j-y,),

(50)

where ye is the a-tube rapidity (see eq. (34)), x=pJmN, (51) pll and pl are the longitudinal

and transversal (with respect to the beam axis z) components of the particle momentum, respectively. In the r.h.s. of eq. (50) we transfer from momentum p to new kinematic variables: the dimensionless transversal momentum x and the longitudinal rapidity y (i). The use of these variables provides the simplest way of transforming from one reference frame @another. The invariant momentum distribution of baryons in the a-tube is obtained by multiplying the value E $“fbf’ by the tube volume in its rest frame dV,. Performing the summation over tubes in the overlap region S(b), we get the following expression for the momentum distribution of the i-particles from the a-nucleus: E(i)

d3NE’ _ d3p

d V, Ebj”j-‘,” = n, I

exp{(rni-E$))/T,}

d’s Z,(s)Eh(” I S(b)

K(T,)

*

(52) The composite particle yield is calculated within the framework of the coalescence model 27) assuming that the light-fragment formation takes place at the stage when the nucleon momentum distribution has already been formed. The momentum distribution functions of primordial protons (marked by a tilde in the following) are determined as FCpj -=d3fiJb4) 2, d’p

A,

d3N”’ d+L&d3q

1 4~~0

E’A’(q)E’p’-q*p_E mA

(53) The 2,/A, factor represents the proton-to-mass number ratio in the initial nucleus. The second term in the square brackets corresponds to nucleons coming from the 227 MeV/ c are respectively the energy A-particle decay ‘2S4’),A+N+~.Eoandpo= and momentum of the escaping nucleon in the A-isobar rest frame. The invariant cross section of the primordial protons is obtained by integration over impact parameters b (b < b,,, = R, + R,): (54)

According to the coalescence model the invariant cross section of fragments with

731

Yu.B. Ivanov et at. f Pa~iaf-tmnspar~cy

mass number A, charge Z and momentum PA is determined by the Ath power of the proton cross section of the same momentum per nucleon:

In”

u'A'(pA)

=

(59

Gld~;:?(~,/A)l~ ,

where C, is the coalescence coefficient independent of p,+ The invariant cross section of the observed free protons a!$ is related to &&! as follows:

(A) decrease rapidly with increasing A, it is sufhcient to Since the cross sections qnv take account of the deuteron, triton and 3He cont~butions, only. In actual ealculalions the experimental coefficients CA were used 6’42)(see table 1). The conventional coalescence model should be modified for calculation of secondary particle spectra in central heavy-ion collisions. Experimental data on coalescence coefficients are absent for this case. To solve this problem, we assume that the power relation between the proton and composite particle momentum distributions takes place at each fixed value of the impact parameter b, i.e. in each individual event. With this modification the invariant cross section of the composite particle A is (A)=

uinv

&b

C CY=p,t

&N(A) E’&__+

dPA

(57)

’

where E(A)

d3NLAP’ ~=~_(~)~-~[E(p)y]~,

p=p~/A.

(58)

The modified coalescence coefficients CA are assumed to be b-independent adjustable parameters. They differ from CA of eq. (55) and even have a different dimensionality. The (Np/Z,)A-Z factor takes account of different neutron-proton ratios in projectile and target. The relation between the primordial and observed proton distribution is given by eq. (56) with the replacement of crf,$ and CA by ECp)d3N’P’/d3p and cA( N,JZa)A-Z, respectively. The representation of the cross section in the form of eqs. (57) and (58) makes it possible to simulate space-correlation effects. It is clear that for composite-particle formation both momenta and positions of coalescent nucleons are required to be close. In the modified version of the model no coalescence between nucleons from space-separated flows arises. Moreover, this approach eliminates the nonphysical interference between different events. As calculations show, the “differential” version of the coalescence model, eqs. (57) and (58), brings about a similar and sometimes even better reproduction of the experimental spectra as compared to the “integral*’ version, eq. (55). The coefficients CA deduced from a comparison with the experimental data are given in table 1.

732

Yu.3.

lfxznou

et al. f Furtial-~ra~~~~cy

TABLE The

%+A,

Ebb

R,

4

[GoV/nucleon]

[fm]

[fml

{ ;I;

::;

:3;

20Ne + *‘Ne 4oAr+40Ar =Ne+2Y% “Ar+“*Pb 2,s”+238u

“)

model

1

parameters

c,

c.3 [GeV*/~

mb’]

1.6 x IO-’ ,,4x,O_~

0.8 0.8

4.7 3.1

4.7 8.0

0.8

4.7

8.0

3 x to-6

0.8

8.3

8.3

4x10-b

8 x 10-6 4 x 10-6

[Gee/@.

C3°C mb”]

2.1 x10-”

[Gep/c6.

c2 mb’]

c

[GeV2/c3]

[Gev/?]

I.Jxlo-’

1.6~10-‘~

IO_‘0

10-10

5X10-” 1.2X10-”

SXW” 6XW’Z

1.4x 10-z

SXlO-‘z

l.4xlo-2

9XIO_‘Z IO-”

8 x1o-S

6x IO-”

“f For the 238U nuclei BII effective radius is employed exceeding 20*P%case. The coalescence as for the “Art

the geometrical one (R = 1.17A lf3 fm) by the fame value BS in the coefficients are evaluated by means of eqs. (29) and (30) of ref. 6, by using the fame coalwzence momenta

KC1 reaction.

5. Numerical results and discussion Numerical calculations were performed for the reactions Ne + Ne, Ar + Ar, Ne + Pb and Ar+ Pb at beam energies of 0.8 and 2.1 GeV/nucleon, since at present most complete experimental data are available on these reactions 6*43*44). In anticipation of future experiments, the reaction U+U at 0.8 GeV/nucleon was also considered. The very useful form of data presentation in terms of x and y kinematical variables (see eq. (51)) suggested in ref. ‘) revealed the two-humped structure of invariant di~erential cross sections. The authors of ref. 5, gave the interpretation of this structure in terms of three (slow, moderate and fast) moving sources emitting secondary particles. As has been mentioned above, our interpretation of this structure is based on the partial-transparency concept. We compare our calculations with the experimental data as well as with predictions of the firestreak model, this being a limiting case of our model at hd = 0. Figs. 2,3 represent the proton and triton inclusive spectra for the Ar+ Ar collision at EIab= 0.8 GeV/nucleon. As can be seen, the model describes the data fairly well, upon the whole, and,‘which is of primary importance, it properly reproduces the two-humped structure at low x. The value Ad= 8 fm was used in the calculations. On the contrary, the firestreak model results in structureless dist~butions at any x, with a maximum at the centre-of-mass rapidity. The two-humped structure is particularly pronounced in the triton spectrum. Within the scope of the coalescence model it has a simple explanation, i.e. the peculiarities become more pronounced because the triton spectra are proportional to the proton spectra cubed. Though the x - y representation has certain advantages, more conventional variables, i.e. the momentum and the emission angle in the laboratory frame, are widely used. The invariant proton cross section in terms of these variables is shown in fig. 4 (the reaction is the same as in fig. 2). Evidently here the degree of agreement with the experimental data appears to be the same as in fig. 2, though in fig. 4 the pa~ial-transparency effects are not so pronounced.

733

Yu. B. Iuanov et al. / Partial-transparency

Fig. 2. The proton inclusive invariant cross sections in the rapidity representation (here and below given in mb * c3/GeV2) for the 40Ar+40Ar collision at I&,, = 0.8 GeV/nucleon. The dash line corresponds to Ad=O; the full line, to Id= 8 fm; the open and full circles, to the experimental data on the 4oAr+ KC1 reaction 5).

Our attention

is attracted

by a systematic

overestimation,

as compared

with the

experimental data, of the particle yield in the tail regions. We believe it occurs due to employing the DF of eq. (43, which does not constrain high single-particle energies. Since the particle number in each tube is not Iarge (not more than 8-9 for the heaviest nuclei), the restriction on the maximum allowed single-particle energy due to the total energy and momentum conservation becomes important. This constraint

may be incorporated

0

by employing

the microcanonical

0.4

0.8

I Yo/2

1.2 I

Yo

Fig. 3. The same as in fig. 2, but for tritons.

distribution

Yt.3,

instead

Yu.B. Zvanov et al. / Partial-transparency

734

6

IltV

IO4

Fig. 4. The proton inclusive invariant cross sections in the conventional representation (for the 4oAr+40Ar The full line corresponds to the calculation disregarding the collision) at E,,, = 0 .8 GeV/nucleon. phase-space constraints and employing A, = 8 fm; the dash line, to the calculation including the phasespace constraints according to eq. (59) and employing the same A,, . the open and full circles, to the experimental data 6).

of the Gibbs

distribution used. Calculations of this “phase-space model” type were calculations, we use carried out in a number of works 2S45).To avoid cumbersome a simplified and, thus, a more rough procedure for estimating this effect. That is, we superimpose a condition that the emitted nucleon energy within the tube rest frame should be not higher than the total tube excitation energy: E &‘N’-mN<[&(T,)-mN]Na,

a=p,t,

(59)

where N, = n&a,, is the mean baryon number in the a-tube, and E( T,) is given by eq. (49). The contribution of particles with higher energies is neglected. They have little effect on the normalization. It is clear that in the real situation particle redistribution over the energies occurs resulting in an enhancement of the soft part of the spectrum (in the rest frame of the tube). The effect of the phase-space constraints is illustrated in fig. 4; it changes the spectrum in the right way. Moreover, by using the condition of eq. (59) we even overestimate this effect. If not indicated otherwise, the results of the calculations presented below are given without employing restriction (59). Fig. 5 represents the triton rapidity spectra in the Ne + Ne collisions at J?&, = 0.8 GeV/nucleon. The two-humped structure is more pronounced in this case as

Yff.B. Zuanou et id. / Partial-transparency

“Ne

+20Ne-t 0.4

+X 0.8

x*2

f

Y&

h

Fig. 5. The triton inclusive invariant cross sections for the 20Ne+20Ne collisions at J&,= 0.8 GeV/nucleon. The dash, dotted, full and dash-dotted lines correspond to Ad= 0, 6, 8 and 10 fm, respectively; the open and full circles, to the experimental data for the “Ne+ NaF reaction ‘1.

compared reaction

to the Ar-nuclei collisions. Therefore, was performed

to demonstrate

a series of calculations for this the sensitivity of the spectra to the single

model parameter A,-,.Comparison with the experimental data indicates X4 = 8 fm is preferable. This value is in agreement with the estimation

that the value based on the

experimental nucleon-nucleon cross section and was employed for describing all the reactions at EiE,,= 0,8 GeV/nucleon. The variation of R, within the limits of -20% brings about noticeable changes of inclusive cross sections near the side maxima at low X. It partially hides the transparency effects. ‘The use of uniform nucleon density distributions (with zero diffuseness~ does not allow us to reproduce the side maxima in the rapidity distributions at the same Ad. On the other hand, and this is important, the firestreak model does not produce the two-humped structure at any reasonable choice of the R,. The theoretical predictions for the U + U collisions are presented in fig. 6. Figs. 3 and 5, together with fig. 6, illustrate the variation of the rapidity distributions when the mass number of the colliding nuclei increases from Ne to U. As is clear from general considerations, the increase in the nuclear “thickness” should lead to attenuation and, finally, to the disappearance of the incomplete thermalization effects. This is the conclusion to be derived from figs. 3, 5, and 6. Certainly, both

736

Yu. B. Ivanov et al. / RzrriaL~ransparency

Yof2 Yo Fig. 6. The same as in fig. 3, btit for the 238tJ+ 23*U reaction.

the central and peripheral collisions are included in inclusive spectra. The peripheral events are characterized by small tube lengths even in the case of collisions of the heaviest ions. Nevertheless, the weight of peripheral events decreases with increase of the mass number. The characteristic impact parameter appears to be 6 = R,, which implies a dominating tube length of the order of R,. Fig. 7 shows contributions of different impact parameters to the t&on inclusive cross section in the Ar+ Ar collisions at Elab= 0.8 GeV/nucleon. As one can see, 6,

'!Ar+'%r-t+X

Fig. 7. Contributions of different impact-parameter intervals into the inclusive triton cross section for the 4oAr+40Ar collision at I&,= 0.8 GeV/nucleon (x =0.2). Curves 1, 2, 3, 4 and 5 correspond to the intervals 0 s 6/b,,, s 1,O C bl b,,, s 0.25,0.25 s b/b,,, C 0.5,0.5 s b/b,,, c 0.75 and 0.75 s b/ b_ s I, respectively; the open circles, to the experimental data ‘).

YuB.

Iva~ov et al.

/

Partial-trnn~~re~~

737

the two-humped structure is enhanced at more peripheral collisions. On the other hand, the selection of central events results in the disappearance of the two-humped structure. Comparison of calculations for near-central collisions with the experimental data is ambiguous due to an uncertainty of the impact-parameter interval selected by the experimental procedure. In ref. “) the central Art- KC1 and Ar+ Pb collisions at Era,,= 0.8 GeV/nucleon were selected by detecting the high multiplicity events. In fig. 8 our results for central Ar+Ar collisions are presented, where events are considered to be central on condition that &~0.4b,,,= 3.8 fm. The calculation reproduces the experimental data at the proton energy E = 200 MeV fairly well. At the higher proton energies a reasonable agreement is achieved only in the intermediate-angle range. The overestimation at higher and lower c.m. angles is probably due to a disregard of the phase-space constraints.

@Ar + ‘“Ar-

I

0

p +x

I

60

I

,

120

4

I

180 0,dq

Fig. 8. The proton angular distributions for the 4oAr+ 4oAr collisions at E,ab= 0.8 GeV/nucleon. E and 6 are the kinetic energy and emission angle of protons in the reference frame moving at the rapidity y, = 0.61. The full and dash lines correspond to b < b,, and b c 0.4b,,, respectively, with A, = 8 fm; the open and full circles, to the inclusive and high-multiplicity-sele~ed data “), respectively.

Let us turn now to asymmetric nuclear collisions. Fig. 9 presents the triton rapidity distributions in the Ar+ Pb collisions at Elab = 0.8 GeV/nucleon. The model satisfactorily describes the experimental data. The difference from the firestreak model predictions is reduced in this case. This is quite natural since, on the average, an increase of the target-nucleus thickness raises the degree of equilibration. For comparison, the calculations within the integral and differential versions of the coalescence model are presented. The differential version yields a better agreement with the data, particularly in the mid-rapidity region. Apparently, it happens due to the. elimination of the nonphysical interference of different impact-parameter contributions. Unfo~unately, the experimental data are rather poor near the rapidity

738

Yu.B. Zvanov et al. f Partial-transparency

6inv 40Ar

+208Pb- t + X

103

a.4

0

0.4

0.8

I.2

Ylab t

Yo Fig. 9. The triton inclusive invariant cross sections for the *Ar+ “‘Pb collision at E,,, = 0.8 GeV/nucleon. The dash line represents the firestreak-model predictions (Ad = 0) employing the integral version of coalescence model; the full and dash-dotted lines, the integral and differential versions of coalescence model, respectively, with Ad = 8 fm; the open and full circles, the experimental data 5).

of the incident

nucleus,

where the calculated

curves are most sensitive

to the choice

of the model parameter Ad (it requires measurements at small angles). The model evidently underestimates the triton yield near the maximum at y = 0.2. It probably indicates the appearance of collective effects of the hydrodynamical type, such as the target-nucleus explosion in central collisions, which are disregarded in our model. The appearance as Pb. An interpretation hydrodynamical model

of hydrodynamical effects is natural in such a large nucleus of these rapidity distributions within the three-dimensional is suggested in ref. “).

The inclusive and high-multiplicity selected angular distributions of protons in the Ar+Pb collision at Erai,= 0.8 GeV/nucleon are shown in fig. 10. In the same way as in fig. 9 the high-multiplicity data are reasonably reproduced here if the central collisions are defined by the condition b d 0.4b,,, = 5.1 fm. It would be interesting to analyze the data at energies Elab lower and higher than considered so far. In the former case, one should expect a better equilibration due to the reduction of Ad, resulting from an increase of the magnitude and a decrease of the angular anisotropy of the nucleon-nucleon cross section. On the other hand, at sufficiently low energies Pauli blocking becomes important, which increases Ad

739

Yu.B. fvanou et al / Partial-transparency I

_

40Ar+m Pb-p+X

IO5 \ IO4 L

LO3

102

I

I

I

I

30

60

90

120

Fig. 10. The same as in fig. 8, but for the 40Ar+Zo*Pb

bhw 1,

"Ne

+

*%k-t

+

1

1

150

180

collision,

y* = 0.48.

9,deq.

X

Id

xc+ /

IO0

IO-

0.4

1

0

0.4

I.2

0.8 t

I.6

2 .o t

Yo

LJ

YO/2 Fig. 1 I. The triton inclusive invariant cross sections for the 20Ne+Z’Ne collisions at Elab= 2. I GeV/nucleon. The dash and full lines correspond to A, = 0 and 14 fm, respectively; the open and full circles, to the experimental data 43).

740

YuB. Ivanov et al. / Partial-transparency

[ref. “)I_ The absence of detailed experimental info~ation does not allow one to say how important the transparency effects are in this case, The growth of the incident energy above 1 GeV/nucleon should be accompanied by an enhancement of two-flow inequilibrium, and, consequently, by a more pronounced two-humped structure. This is due to the increase of the initial rapidity difference, as well as to the growth of Ad as a result of increasing angular anisotropy of cross section. Moreover, the inelastic processes become more important, which contribute mostly to the mid-rapidity source. We believe that proper consideration of the mid-rapidity source is crucial for an adequate treatment of pion spectra. But the two-flow approximation seems to be sufficiently good for the description of baryon distributions. The leading-particle effect, which is well known in hadron-nucleus collisions at high energies “), is an indirect justification of this conclusion. In fig. 11 the inclusive triton spectra for Nef Ne collision at Elab = 2.1 GeV/nucleon are presented. To obtain a satisfactory description of the data, we have to employ the value Ad= 14 fm. In addition, the maximal admissible value of R, (R,,= 4.1 fm) has to be taken. The increase of R, probably simulates the excitation of a wider region of spectators. It is worth mentioning that the firestreakmodel predictions strongly disagree with the data in this case. It should be noticed that the deceleration-length parameter Ad implies a value somewhat averaged over the nucleon relative energies 15;,, in the range from zero up to J$,,,. The reason why Ad deduced from the data is Elab dependent is that the nucleon-nucleon cross section is energy dependent. For instance, at low energies, E,,, % 0.4 GeV, the 1f Erel law is approximately valid. Due to the lack of detailed experimental information for various beam energies the Eras dependence of Ad cannot presently be extracted from the data.

6. Conclusion In conclusion we summarize the main results of the work. (i) Our model based on the kinetic equation reproduces the available experimental data fairly well within the wide range of secondary-particle energies and emission angles, as well as for a large variety of colliding nuclei. (ii) It is shown that the experimentally-observed two-humped structure of the secondary-particle rapidity distributions cannot be reproduced by the conventional firestreak model and apparently requires for its explanation the inclusion of partialtransparency effects. The experimental data analysis gives an effective deceleration length Ad= 8 fm for Elab = 0.8 GeV/nucleon and A,+= 14 fm for Elab = 2.1 GeV/nucleon. (iii) In agreement with the experimental data our model predicts that the twohumped structure of the rapidity distributions (a) is reduced if central collisions are selected and when transferring to heavier colliding nuclei, and‘(b) is enhanced

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with the increase of the bombarding energy and the mass of secondary composite particles. (iv) The fast and slow sources in terms of ref. ‘) are formed preferably in the peripheral nuclear collisions, whereas the moderate source is associated with the central collisions. It would be interesting to search for the correlation between particles from fast and slow sources. In view of our results several proposals for future experiments may be put forward, which will help to explore better partial-transparency effects: (a) to measure composite-particle rapidity and angular distributions separately in central and peripheral collisions; (b) to extend the measurements into the regions of low x (small perpendicular momenta) and small angles (in the case of a nonsymmetric projectile-target combination) to which the present model is most sensitive; (c) to observe the correlation of particles emitted from slow and fast sources and the anticorrelation of particles from slow (fast) and moderate sources. Naturally, the data on the heaviest ions like U will be of great interest, since in this case one expects an increasing degree of thermalization (minimal transparency) and more pronounced collective phenomena. The authors gratefully acknowledge useful discussions with V.I. Man’ko. One of us (I.N.M.) is grateful to J. Bondorf and J. Zimanyi for their interest in this work as well as to the Niels Bohr Institute and Nordita for the kind hospitality. We thank N.N. Serebryakov for the help in the preparation of the manuscript.

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