Particle emission in light and heavy ion reactions

Nuclear Physics A400(1983)173c-190~. @North-HollandPublishing Co.,Amsterdam Not to be reproduced by photoprint or microfdm without written permission from the publisher.

173c

PARTICLE EMISSION IN LIGHT AND HEAVY ION REACTIONS

V.D.TONEEV and K.K.GUDIMA

Joint Institute for Nuclear Research, Laboratory of Theoretical Physics, Dubna, USSR

Abstract: Physical effects arising from heavy ion-induced reactions as opposed to light ion-induced reactions are discussed in the framework of the cascade model. Our approach is shown to be suitable for the study of collective effects due to nuclear matter compression.

1. Introduction

High-energy heavy-ion collisions provided a unique opportunity to study the behaviour of a hot and dense nuclear matter. For such extreme conditions a number of new phenomena like phase transitions 6f nuclear matter into the pion condensate, density isomers, and quark matter are theoretically predicted. All these phenomena deal with collective behaviour of the nuclear matter. However, if such collective phenomena do exist, they will depend very much on dynamical evolution of the nuclear collision process. Moreover, the collective phenomena will be masked and superimposed by the background produced by quasifree particle-particle collisions. In aiming for a search of signals of the collective interaction, we shall consider in this report nucleon-nucleus and nucleus-nucleus collisions on the same microscopic footing. Such a comparative analysis will allow us, on the one hand, to make some general predictions for heavy-ion collisions and, on the other hand, to more clearly reveal physical effects which set in while passing from nucleons to heavier projec‘tiles. The analysis is based on the intranuclear cascade model I-2) which has turned out to be very successful in the case of hadron-nucleus reactionsj).

2. Basic equations

In the first fast stage of nuclear interaction an intranuclear cascade develops. Mathematically, it is described by the relativistic Boltzmann equation for a one-particle distribution function fA (x,p* ) = fA of a mixture of gases

where D,,ee is the collisio7n term, X~ and pp are the four- dimensional coordinate and momentum of a particle. If a nucleus A impinges on a nucleus B, three kinds of gas are considered: projectile spectators (index A ), target spectators (B) and cascade particles or participants (C). An important simplification of cascade equations comes from the neglect of the interaction between particles of the same kind

174c V.D. TONEEV, K.K. GUDIMA

(I)

and the relative velocity, vzep , are related by 9~=cffII/ (pacr, ,D04'). Properties of hadron-nucleon collisions enter into the system (1) des- cribins the two-nucleus interation through the differential distributions w and total cross section G*,,* , the latter being cor- rected for the Pauli exclusion principle. With given initial conditions, solving the system (1) for each value of the impact parameter b (f(x,p) - f(?,c,t;b)) , we can trace the evolution of the nuclear system and construct any measurable quantities, e.g., the inclusive particle spectrum

dG A+S-c-

=jd3rJ2nWtB)BdBfCIF,p’,t=t,,,,; 8) (2) d’P

where W(b) represents the probability for two nuclei to interact at the impact parameter b. Upon completing the cascade stage

(t _ tC,SC ) there remain excited residual nuclei. At a subsequent more slow stage of the interaction, particles can be emitted both from the equilibrium and non-equilibrium state. We have taken into account the pre-equilibrium emission effects within the exciton model based on the master-equation4)

PARTICLE EMISSION 175c

where P, (E,t)is the probability of finding the system at time t in a state described by the exciton number n (i.e. the number of the particle: above plus the.holes below the Fermi level) and excitation energy E , J? (n) and 21 (T,n) are transition rates into states with n'=n+2 and into continuum state via emission of a particle "j" with a kinetic energy T. The initial state for the master-equation (3) is found from the functions f" and fa resulting from the pre- cedent cascade calculations. When the exciton number exceeds its equilibrium value, n> neg , we arrive *at the conventional theory of the equilibrium decay (evaporation model).

3. Depletion effect

The structure analysis of (1) and comparison with the well- known experimental data on hadron-nucleus interactions allow one to make certain predictions on the importance of the depletion of spectator nucleons for heavy ion collisions.

In the limit of hadron reactions when the nucleus A is replaced by a nucleon, the first equation in (I) is absent, and in the others the terms with the factor fA should be omitted. Using conventional three-dimensional variables, x,p - r',?,t, we may now rewrite the system (1) as follows

&+~v'if8=-f8gC

~&+~v',f"=-f'p,

(4b)

where averages over the distribution functions are determined in the standard manner

with the normalization to the particle number density

The integro-differential equation (4b) can be reduced to the integral equation

176~ V.D. TONEEV, K.K. GUDIMA

In this integral form the physical meaning of the cascade model is most clearly manifested: the probability to find a fast particle is governed by all the preceding collisions taken with the exponential absorption factor of survival.

In a naive cascade approach (in the sense of Goldberger's mo- de15)),one neglects the time dependence of the nucleon number density of the target nucleus, i.e.

The joint solution of (4a) and (4b) takes into account the depletion of the nucleon density of the target nucleus in the course of development of the intranuclear cascade. As follows from (4a) the nucleon density decreases exponentially, and this decrease may essentially influence the dynamics of subsequent collisions in the case of high density of cascade particles (i.e. at high bombarding energies) or for targets with a small nucleon number. The depletion leads to saturation of the beam-energy dependence for the knocked- out-nucleon number, excitation energy and transfer momentum of a residual nucleus (the limiting fragmentation of the target nucleus). Physically these phenomena result from the finite number of nucleons in an interacting nuclear system.

For hadron-nucleus reactions such a limiting fragmentation is observed experimentally, and onset of this regime is fairly well described within the cascade mode11r6 ). As follows from photoemul- sion data, the number of gray and black tracks does not practically depend on energy above - 4 GeVl). This boundary energy decreases with the target-mass number as fast as Bv3 , amounts to about 0.8 GeV l).

for a carbon target it

If we turn back to heavy-ion collisions, the solution of the cascade system (I) will result in limiting fragmentation both of the target and projectile. The boundary energy of the limiting fragmentation range will be reached earlier than that in the hadron- -nucleus collisions by about a factor of A'j3 (B%) for fragmentation of the target (projectile). Sandoval et al.') estimated the energy dependence of the number of protons, Q, involved into central Ar+KCl collisions

Q= ntot -2n,-- m,,,~ tazj I

where ntot and n,- are the number of all charged particles and negative pions, npw is the number of leading fragments tra- velling with the projectile velocity in a 4' forward cone, and ntaY is the number of positive tracks observed with p i 200 MeV/c. It turns out that for central collisions Qz 28 and is practically independent of the beam energy in the range T,+ 0.9 GeV/nucl.'), whereas for the p+ Ar reaction the boundary energy equals approxi- mately 2 GeV 3).

4. Coalescence effect

When one moves from nucleons to heavier projectiles the density of cascade particles increases and assumption that after time t,,S, the cascade particles do not interact (see formula (2)) becomes still


l”~‘oe- -203 T lMeVl

7--.--r

IO' Ne+U - p* X i025GeV/nucl~

J 303

1

0 100 203 3co 0 la, 2M) 301

101 I+.“- p+x ( 2,i Gev/nucl )

'A

k v-t

10' .

*\

,oo ** 11OO

0.' . %

10' ~ 7001.10-'1

";.

s\ ;. b$, 1 130%0',

.

:

10'

1Ci'

9oo,.,0.,)/ * ,;r'

0 100 200 300 0 100 200 3K

T IMeVi

Fig. 1. Inclusive spectra of particles emitted in the Ne+U collisions at d'fferent projectile energies.

8 Experimental points are taken from

ref. ). Histograms are calculated within the cascade model with effects of the coalescence and pre-equilibrium emission.

17% V.D. TONEEV, K.K. GUDIMA

less justified. The particle interaction in a final state can give rise to the coalescence of nucleons into composites. We shall take into account this effect through the additional assumption that all the cascade nucleons,havinq the relative momenta in momentum space smaller than pc and correct isotopic content,form an appropriate composite particle. This means that the formation probability for, e-9., a deuteron is

where the particle density in momentum space is related to the one- particle distribution function by

(8)

Calculated results of inclusive spectra for protons and composite particles including the nucleon coalescence and pre-equilibrium effects are shown in fig. 1 for the Ne+U reaction in the projectile-energy range from 0.25 to 2.1 GeV/nucl. The coalescence radii were chosen for that reaction at energy T, = 0.4 GeV/nucl. The values are as follows: and pc c4He) = 115 MeV/c. The

pc (d) = 90 MeV/c, pc c3H) = pc c3He) = 108 MeV/c same coalescence radii provide a reason-

able agreement with experiment in the whole energy region under consideration. High-energy composite particles are entirely due to the nucleon coalescence. The deviation of theory from experiment in low- energy pare of spectra comes from a simplified description of the nuclear de-excitation process* ).

The projectile dependence of proton spectra is drawn in fig. 2. A good agreement with experiment for the reactions induced by protons and alphas is not unexpected. For Ne and even more for Ar ions theory underestimates the proton yield in the angular range Q= 50'-70' (i.e. in the region where hydrodynamical effects could be expected), though at larger angles the calculation results are again in a good agreement with measurements.

Hydrodynamical calculations of inclusive spectra were made only for the Ne+U reaction at T, = 0.25 and 0.4 GeV/nucl. gflo) and similar agreement with experiment was achieved that allowed the conclusion on a weak sensitivity of inclusive spectra to the reaction mechanism. As follows from results presented above, such an analysis should be carried out more thoroughly and with a wide set of data.

In fig. 3 we show invariant inclusive spectra measured by Naga- miya et al. 'I) for a symmetric combination of colliding ions. Unlike the experimental result of ref.B), discussed above, the secondaries here are measured inadifferent kinematical region. Nevertheless, in this case the cascade model agrees with experiment as well. The theory predicts a more pronounced peak of quasifree elastic scattering in proton spectra that will result in stronger proton correlations. A satisfactory agreement is also obtained for pion spectra.

*) We have used a "sharp-cut-off approximation" to pass from the cascade to pre-equilibrium stage that results in a somewhat larger yields of low-energy particles and in a gap for the spectrum shape 4).

PARTICLE EMISSION 179~

TlMeVl

T IMeV)

10" I ---

R ‘k o I, - p I x (10~ GcV/nuc~ I

i0’

10”

10'

%P

Fig. 2. Inclusive proton spectra from reactions induced by different ions with energy 1.04 GeV/nucZ. The notation is the same as in Fig.1.

It is of interest that all the above results were established with the same values of coalescence radii pC independently both of the beam energy and tasget-projectile mass combination. This seems to contradict the recent data of Nagamiya et aYLS11) who has shown a strong dependence of the coalescence radius p. on the colliding ion combination. Rowever, the determination procedures af pt and p0 are completely different. The parameter pC defines the effec- tive interaction range of particles in momentum space and composite particles are collected provided the energy-momentum conservation law is fulfilled in each "star". while the parameter p0 is

18Oc V.D. TONEEV, K.K. GUDIMA

lo+-, a, ‘L 10

0 1 2 0 1 2

plGeV/c 1

102.

3o” tx 10-l 1

10'.

\J! t . f

100.

0 0.5 1 1.5 2 2.5 3 35 L

P IGeV/c )

105 ( , I , t ‘ -r-

Ar*KI - ll‘ +x iO.BGev/N

Fig. 3a. Invariant inclusive distributions for protons, deuterons and pions produF?d in the Ar+KCl reaction at T=0.8 GeV/nucl. Points are experiment ), histograms are our calculation results.

1043

, 0 1

2 3

t

5

p G&/Cl

0 1

2

3

L

5

6

P LGeV/c 1

Fig. 3b. Invariant

inclusive

distributions

for camposite particles

produced

in the

Ar+KCL reaction at

T = 0.8 GeV/nucl.

Points are experiment

?I), histograms

are

our

calculated

results.


extracted from the ratio of the invariant inclusive (i.e. summed up over all "stars" and impact parameters) spectrum of the composite particle having A nucleons to the A-th power of the corresponding proton spectrum. To check the consistency of both procedures, we plot in fig. 4 the ratio

for the Ne+NaF reaction at three energies. The agreement with experiment is not worse than in the nuclear firestreak model sup- plied with the chemical-equilibrium hypothesis.

As was first noted bv Siemens and Kaousta12), the deuteron-to- roton ratio, k Rdp' may be-used to estimate the entropy per baryon,

produced in heavy-ion collisions via

3=3.95-h-/ Rdp* (10)

This formula is derived for an ideal classical gas of nucleons assuming R (10) we ha% yJi.7

From cascade calculations of the R values with 5.9 and 6.4 for the Ne+NaF reacti

0.8 and 2.1 GeV/nuc;., accordingly. dl: n at T, =0.4,

The overall trend reproduces the experimental datall), which is not unexpected.

5. Signs of the compression effect

The cascade model is less justified for central collisions than for peripheral ones. Therefore, to search for the most noticeable deviations from experiment which may be related to signs of the compression effect, we turn to analyze the exclusive data selecting events with a small impact parameter. In fig. 5 predictions of the cascade and fluid-dynamical models are compared with data of Gutbrod et a1.14) for the reaction Ne+U (0.4 GeV/nucl.). Due to selection of the events with high multiplicity of charged particles the range of parameters b (2.6 fm was picked out. These experimental conditions were simulated in the cascade calculations as accurately as possible. The contribution of different impact parameters calculated with the cascade model, W(b), was used as a weight function in hydrodynamical calculations10115) - It is to be noted that we used the most elaborate version of the hydrodynamical model taking into account the binding effect, particle "evaporation" from moving fluid cells and cons'

YE! erinq particles of various kinds

by assuming chemical equilibrium ). As follows from results shownin fig. 5 both the approaches

give similar results which agree qualitatively (up to a factor of 2-3) with experiment. It is doubtful whether it may be unambiguously


Ne*NoF 800 McV/nud

l-----+----l

ZlOO MeV/nud.

I

0.5 I.5 0 I.0 2.0 0 LD 2.D I.0 20 3.0 Pd (ceV/cI

Fig. 4. The d/p2 ratio (see the relation (9)) at pd = 2pp for the Ne+NaF collisions at three values of projectile energy. points are from ref:ll).

Experimental Histograms are cascade results with the

effects of coalescence and pre-equilibrium emission. The dashed lines represent predictions of the firestreak model.

EXP CEM NFDM 1

0 30 60 go 120 0 30 60 90 120 150 30 60 90 ml 150

Oldegl

Fig. 5. Angular distributions of particles with kinetic energy per nucleon 12(*),21(~),47(o)and 86WMeV produced in' central (high-multiplicity selected) collisions of neon ions with uranium nuclei at T,= 0.4 GeV/ nucl. The xperiment is from ref.14), hydrodynamical calculations are from ref. I? ). The dashed line represents the cascade calculation by Yariv and Fraenkel 16) and fluid dynamic calculations by the Nix groupg).


interpreted as a manifestation of effects of the hydrodynamical compression. Note should be made that the fluid-dynamic model without "evaporation" effects gives in fact a qualitatively different result (dashed line in fig. 5); however, it strongly contradicts the experiment.

The results of the above comparison seem somewhat unexpected, SO let us compare in more detail the interaction dynamics in the cascade and hydrodynamical models. This is convenient to carry out in terms of the velocity field in combination with the levels of cons- tant density and temperature that provides an insight into the direc- tion, energy and intensity of produced particle emission. The nuclear fluid dynamics (see fig. 6) predicts the suppression of the forward emitted particles at small impact parameters and the "bounce-off" effect at large impact parameters, both the phenomena being treated as a direct manifestation of the nuclear matter compression 10~17).

Knowing the solution of cascade equations (I), fC (?,G,t), and using the averaging prodedure one can estimate macroscopic quantities requireclto construct the velocity diagrams* ). These calculations are also presented in fig. 6. It is seen that the sideward splashing of particles predicted by nuclear Hydrodynamics in head-on collisions and the "bounce-off" effect in non-central interactions are present also in cascade approach though much less pronounced. Therefore, an attempt to differentiate the hydrodynamic-compression effects in central collisions is unfortunately a problem of the quantitative comparison with experiment of macro- and microscopic predictions rather than of the search of qualitatively new effects.

Another interesting group of exclusive data was obtained in the streamer chamber experiment7). The cascade model fails to reproduce quantitatively the energy dependence of the mean multiplicity of negative pions produced in the central Ar+KCl collisions. At the maxi- mal energy measured, the theory predicts dn,->=7.2 that exceeds con- siderably the experimental value <n,_>=5.79+ 0.04 7). Distributions over nR- and the total number of charged particles, ntot , are drawn in fig. 7 both for all inelastic events and for central collisions. It is seen that the theory gives too large multiplicities of particles at small impact parameters though the general shape of the distributions agrees with experiment. It is to be noted that the theoretical distributions are normalized to the reaction total cross section calculated within the cascade model, GR=2.3 b, which exceeds the experimental valueGR=1.9* 0.1 7). Despite these discrepancies the theory reproduces fairly well the correlations between particles produced (fig. 8) and the absolute number of protons, Q, involved into the reaction** ).

* ) The particle number density is given by the relation (6). The average velocity can be found by a relation of the type (5). To connect the particle average energy in the rest system with the temperature at each time moment t, the one-particle distribution function of cascade particles is assumed to be approximated by the relativized Maxwell-Boltzmann distribution 2).

**I In central 7 ollisions at To = 1.81 GeV/nucl. the experimental value Q = 28.02 0.1 ) practically coincides with the calculated one, Q= 27.4.


p_au,w J0.2c

p=01fm-3 2hn

Fig. 6. The velocity field at the time moment t= t,,,, for the reaction Ne+U (0.4 GeV/nucl.) at two values of impact parameter b. Levels of the equal density and temperature are shown, as well. Calculation results are obtained within nuclear fluid dynamics18) (to the right) and cascade (to the left) models.

Undoubtedly, the manifestation of collective interaction effects is more probable in collisions with a small impact parameter. However, discrepancies observed may include a contribution of comparatively trivial effects having no direct relation to the nuclear matter compression. In particular, the straight-line approximation for the relative motion of the colliding ions in our cascade model becomes suspect for heavier ions. Giving up this approximation means that simultaneously with the cascade equations (1) one has to solve a motion equation for ions under the action of a Coulomb poten- tial, VcouL , collisions

and momentum transfer,A 4 , resulting from intranuclear

(11)

186c V.D. TONEEV, K.K. GUDIMA

n tat nn-

7 Total charged particle ("tot ) and negative pions (n,- ) Et kpiicity distributions for the interaction of Ar+KCl at 1.81 9 GeV/nucl. for the inelastic ( l ) and central (0 ) trigger modes. Experimental points are from refL7), histograms are cascade model predictions.

IL

t

40Ar + Kc1 IO - I.8 GeV/nucl.

/: P

c L

. 6

: l

4 8:X, : 2

0 IO 20 30 40

Q 0 2 4 6 8

(n,-)

Fig. 8. Correlations of negative pion average multiplicity, <ns-> , with a number of participant protons, Q, and the <II,-> dependence of the squared dispersion of the 15 -meson multiplicity for a given Q in the Ar+KCl reaction at T,= 1.81 GeV/nucl. Full circles are experimental points from ref. 7). Theoretical. results are obtained within the cascade model. For a more clear representation theoretical points for the quantity &_(< 17,~>) are shifted up by two.


Here R' is a relative distance between nuclei with reduced mass/r , and the summation runs over all intranuclear collisions.

Detailed calculation are in progress. However, it is qualitatively clear that this effect will be important for sufficiently heavy ions and will result in an increase of the average impact parameter and hence in a decrease of the multiplicity of cascade particles. The enlargement of the system (1) by the equation (11) in fact means the inclusion into consideration of the angular momentum conservation law for the whole interacting system. This fixes the reaction plane and will result in two-particle correlations like those predicted by the hydrodynamical "bounce-off" 15).

Sphericity

Fiq. 9. Results of the event-by-event analysis of negative pions produced in the central Ar+KCl collisions at To = 1.81 GeV/nucl. Experimental points are from ref.lg), histograms are cascade calculations.

It is of interest also to carry out measurements at comparatively low energies (T, _ 100 MeV/nucl.) and with very heavy ions when in principle one may observe stars with a large number of tracks but having no charged particles in a narrow forward cone due to the Coulomb-field effect.

In recent years certain hopes in identification of collective effects come from a global analysis. In fig. 9 the comparison with experiment is presented for the collective variable "sphericity"

.S f2

and its orientation angle SsPwL for negative pions produced in the central Ar+KCl collisions at T, = 1.81 GeV/nucl.lg). A noticeable dis- agreement in the BspH -distribution reflects the above mentioned discrepancy in the K--meson multiplicity which is mainly due to low energy pions emitted backward in the c.m.s. of colliding ions.


6. Concluding remarks

From the above considerations it is seen that the cascade approach developed is quite reasonable and allows us to reproduce many features of nucleus-nucleus collisions. Therefore, this approach seems to be rather promising as a basis to analyse, by the diffe- rence method, the collective effects of multi-particle interactions and, in particular, to ascertain the importance of nuclear-matter compression.

It is to be noted that the cascade approach in the modern form- ulation differs essentially from the naive concept considering the cascade mechanism merely as a sum of independent intranuclear cas- cades. Some collective features of the nuclear interaction have been taken into account by effects of the nuclear depletion, nucleon coalescence and pre-equilibrium particle emission. However, of more importance is that the revealed deviations from experiment point the way to further development of this approach. Some of these necessary developments are as follows:

i) Giving up the straight-line approximation for colliding ions; ii) improving the description of the de-excitation stage of re-

sidual nuclei, heavy (the non-equilibrium decay of nuclei with a high angular momentum) and light (a decay of the explosion type);

iii) transport of slow pions through the nuclear matter is an open problem;

iv) estimating the influence of particle-particle interaction during the cascade stage.

If these factors were included, the model predictions would be changed to a c.ertain extent, and a more convincing analysis could be made of signs of collective effects.

References

1. K.K.Gudima and V.D.Toneev, Yad.Fyz. 27 (1978) 658. 2. K.K.Gudima, H.Iwe and V.D.Toneev, J.%ys. 65 (1979) 229. 3. V.S.Barashenkov and V.D.Toneev, Interactionof high energy

particles and atomic nuclei with nuclei (Atomizdat, Moscow, 1972), in Russian.

4. K.K.Gudima, S.G.Mashnik and V.D.Toneev, JINR communication P2-80-774; P2-80-777, Dubna, 1980.

5. M.L.Goldberger, Phys.Rev. 74 (1948) 1268. 6. V.S.Barashenkov, A.S.Iljinov and V.D.Toneev, Yad.Fyz.

13 (1971) 743. 7. KSandoval, R.Stock, H.E.Stelzer, R.E.Renfordt, J.W.Harris,

J.P.Branniggan, J.V.Geagh, L.J.Resenberg, L.S.Schroeder and K.L.Wolf, Phys.Rev.Lett. 45 (1980) 874.

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9. A.A.Amsden, F.H.Harlow, J.R.Nix, Phys.Rev. Cl5 (1977) 2059; Cl8 (1978) 1293.

10. G.Buchwald,L.P.Csernai, J.A.Maruhn, WLGreiner and H.St&ker. Phys.Rev. C24 (1981) 135.

11. S.Nagamiya, M.-C.Lemaire, E*.Moeller, S.Schnetaer, S.Shapiro, H.Steiner and I.Tanihata, Phys.Rev. G24 (1981) 971.

12. P.J.Siemens and J.I.Kapusta, Phys.Rev.Lett. 43 (1979) 1489; - 43 (1979) 1690. -


13. H-Stocker, Lawrence Berkeley Laboratory preprint :J012302 (1981).

14. R-Stock, H.H.Gutbrod, W.G.Meyer, A.M.Poskanzer, A.Sandoval, J.Gosset, C.H.King, H.King, Ch.Lucker, Nguen Van Sen, G.D. Westfall and K.L.Wolf, Phys.Rev.Lett. 44 (1980) 1243.

15. L.P.Csernai, H.St&ker, P.R.Subramaniac G.Buchwald, G.Gxaebner, A.Rosenhauer, J.A.Maruhn and W.Greiner, Report of Central Research Institute for Physics, KFKI-1982-31, Budapest, 1982.

16. Y.Yariv and Z.Fraenkel, Phys.Rev. C20 (1979) 227; C24 (1981) 488.

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-- J.Maruhn and W.Greiner, Phys.Rev.Lett. 44

(1980) 725. 19. R.Stock, Lawrence Berkeley Laboratory preprint No12884 (1981).

Documents

Particle emission in light and heavy ion reactions