A cooperative mechanism of subthreshold pion production

Nuclear

Physics

@ North-Holland

A426 (1984) 606-624 Publishing

Company

A COOPERATIVE

MECHANISM OF SUBTHRESHOLD PION PRODUCTION R. SHYAM

and

J. KNOLL

GSI, Postfach 110541, Planckstrasse 1, D-6100 Darmstadt 11, Federal Republic of Germany Received

16 April

1984

Abetract: We present a statistical model for studying pion production at subthreshold beam energies (<290 MeVIA). This model involves the cooperative action of several of the target and projectile nucleons in the pion-production process. We also consider the formation of fragments in the final channel alongside the produced pion. Calculations performed within the model provide a good overall description of the experimental data over a wide range of beam energies and masses of the participating nuclei. Fragment formation in the final channel is seen to be vital for understanding the experimental data.

1. Introduction

One of the interesting aspects of heavy-ion collisions at intermediate energies is the observation of a significant pion-production cross section at the so-called “subthreshold beam energies” (i.e. below 290 MeV/A) IA). These pions open up a kinematical domain which is not accessible in free nucleon-nucleon (NN) collisions. Theoretical attempts to understand their production mechanism range from the nucleon-nucleon single-collision (NNSC) picture ‘*‘) [suggested even prior to the laboratory detection of the pion 9)], where the intrinsic (Fermi) motion of the nucleons in the initial states of the two colliding nuclei provides the necessary kinetic energy, to collective processes such as the mean-field approach lo), decay from the compound nucleus ‘I), and pionic bremsstrahlung I’). In an earlier study it has already been shown “) that the NNSC mechanism with a realistic shell-model prescription to describe the intrinsic nucleonic motion in the initial states of the colliding nuclei is insufficient to explain the absolute yields as well as the shapes of the pion spectra. In the mean-field approach, the dynamics of the collision is described by the time-dependent Hat-tree-Fock theory, and the pion production is treated in a first-order approximation by the so-called one-nucleon or two-nucleon mechanisms in the microscopic time-dependent orbits. The mechanism of pionic bremsstrahlung is even more collective in nature as it only involves the ground-state densities of the two colliding nuclei. However, this depends sensitively on the assumed parameter related to the deceleration of the nuclei during the collision. The collective processes mentioned above treat the one-body part of the nucleusnucleus interaction rather rigorously, i.e. they account for that part of all interactions 606

R. Shyam, J. Knoll / Cooperative

mechanism

607

which can be included in a one-body field, while the residual two-body interaction part is treated at best only perturbatively. In this paper we present a model where contrary

to the collective

picture,

one takes care of the residual

interaction

part of

the nuclear collision dynamics and the more smooth one-body part is ignored, an approach which becomes more and more valid with increasing beam energy. In the sense of a quantum multiple-collision picture our model considers the genuine cooperation among several nucleons which allows the pooling of their energies to produce a pion. A further cooperative effect results from the formation of fragments of various masses in the final channel alongside the produced pion. In order to simplify calculations of the spectral forms we invoke the assumption of the equal occupation of the available phase-space. In the next section we present the details of our model. In sect. 3, results of the calculations performed within the model are compared with the experimental data. Finally our conclusions are presented in sect. 4.

2. The

model

The starting point of our model is the general observation that in a quantummechanical multiple-collision picture off-shell collisions allow a by far more flexible sharing of the available energy than would be possible by sequential on-shell scattering (as in a cascade model). Such a genuine cooperative mechanism allows the pooling of the energies of all those nucleons which are in mutual interactioncontact during the collision. Faced with such a picture the whole collision process can be visualised in terms of a grouping of all the nucleons into what we shall call clusters, each cluster containing all those target and projectile nucleons which are in interaction-contact with one another during the collision process. To be specific on this point, we do not think so much of preformed clusters in the sense of deuteron

or a-particle correlations in the initial nuclear configurations, but rather of clusters formed dynamically through interactions occurring during the collision process. As a consequence one-body observables such as the inclusive cross section for observing a particle

y can be expressed

from all the clusters

as an incoherent

sum over the contributions

arising

13*14): sJd3aldp:)

= C a,,(M MN

W%.J(PT).

(2.1)

labels M and N represent the numbers of the projectile and target nucleons, respectively, in each contributing cluster. In a diagrammatic expansion of the whole collision dynamics a cluster contribution (M, N) would comprise the absolute square of the coherent sum of all those diagrams which contain a connected piece with M projectile and N target legs. Since different clusters reach different final channels we add them incoherently in eq. (2.1). The yield of each cluster is factorised into a formation cross section u,& M, N), and a properly normalised probability distribution FLN. While formation cross sections specify the occurrence of a given The

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R. Shymn, J. Knoll / Cooperatiw

mechanism

cluster and thus carry the signature of the incident channel, the spectra &,, give the partial flux that goes into the observed channel out of the cluster (M, N), i.e. to the observe particle y with energy E, and momentum jr,_ Note that eq. (2.1) as such is quite general and accommodates quite a variety of multiple-collision approaches. Different prescriptions for calculating a,, and F& lead to different models, e.g. the NNSC or thermal models ‘5*16).In the present study we employ the nuclear phase-space model as already explained in ref. ‘3*17).However, we incorporate into it two essential extensions. One concerns the specification of the formation cross sections. Here we leave the straight-line communication limit [rows-onrows ‘“)I which was quite successful at higher energies and turn to the communication as resulting from a 3-dimensional cascade 19). The second extension concerns the final states, which here also include the possible formation of composite light nuclei besides free nucleons and the considered pion. Both extensions though less important at higher energies become an essential part of the model due to the severe kinematical constraints imposed by the subthreshold condition. Therefore, we shall devote the two following subsections to an explanation of the details. 2.1. FORMATION

CROSS SECTIONS

At higher energies the straight-line communication limit was quite a successful tool to estimate the formation cross sections for the following reason. It gave a proper account of the occurrence of the small clusters providing the non-equilibrium component of the reaction, while it largely underestimated the large-cluster formation. As the latter showed a limiting behaviour (bulk limit), a redistribution of the medium-size clusters in favour of larger ones would not have altered the resulting cross sections. For kinematical reasons the subthreshold production mechanism becomes increasingly important with growing cluster sizes. One requires a pooling of energy in order to surmount the threshold c.m. energy of the pion mass. This demands a careful estimate of the formation probability of the large clusters. Within the kinetic collision-dominated regime which we are considering, the presently available optimal procedure to obtain formation cross sections would be to extract them from a 3-dimensional cascade calculation, the idea being that the cascade dynamics contains the essential ingredients to determine the interaction-contact among the constituents of the reaction, the nucleons: these are the geometry of the initial nuclear configurations and the size of the NN cross section. Such a cluster analysis of the cascade has been performed, although at higher energies, and we like to take advantage of the experience obtained in that study 14) and transfer this to our energy regime via two assumptions. The first and essential assumption is that the connectedness, i.e. the occurrence of various cluster sizes, does not change with bombarding energy (this allows us to use in our energy regime results of the cluster analysis of the cascade performed at higher beam energy). At first glance one might wonder as to how far this assumption

R Shyam, J. Knoll / Coopraiiw

609

mechanism

is justified because the measured NN cross section increases steadily (going above 120 mb) as the beam energy is lowered. However, one should keep in mind that the most effective sharing of energy comes about through off-shell scattering processes, i.e. the near-zone behaviour of the corresponding wave functions. The highmomentum components of these wave functions (which are more relevant for us) are particularly determined by the hard-core part of the NN force. This, however, is essentially given by the geometry with a cross section of about 10 to 15 mb attributed to it. This is about the value which was employed in the cascade cluster analysis of ref. 14). Viewed in this way the formation cross section depends only on the geometry and not on the beam energy. The second assumption concerns with the calculation of the formation cross sections. We hesitated repeating the respective cascade analyses for all the cases. Rather, for a handy treatment we preferred an analytic expression that reproduces at least the gross behaviour of the cascade formation cross sections. Interested only in larger clusters (M + N > 4), we found that the old rows-on-rows formula ‘3S’7), however, used with an enlarged effective NN cross section is capable of reproducing the growth to larger clusters encountered in the cascade. A resonable fit to the formation cross section of ref. 14) was obtained by a value of a&, = 120 mb. This value was kept constant throughout in all the calculations discussed below. For completeness we cite the analytic expressions used: aA,=

I

d2b d2spd2sT S2(b +s,-s-r)PA(M,

s,)P,(N,

+)/o$,

.

(2.2)

The Poisson distributions pA(“,

NA(

s,

=

(NA(S))M

s)

=

a;,., I-

exp

-m

PA(s,

(-NA(s))/M! Z)

dz,

,

(2.3)

(2.4)

account for the fluctuations in the collision number. We take the nuclear densities PA and pB in a standard parametrization as described in refs. 2S22). 2.2. SPECTRA

The spectra FLN( p,) are a result of the underlying dynamics. For the kinematical regime of interest we are sure that a cascade dynamics, i.e. a sequence of on-shell scattering processes, will certainly not provide enough collectivity to ensure the required threshold energy. As it has been shown “) that the production rates are not sufficient already for the first NN collisions, which with high probability are the most energetic ones in a cascade dynamics, one cannot expect the production to result from the subsequent NN collisions. Thus we have to consider the cooperative effects which are beyond the scope of a traditional cascade treatment. This should

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R Shyam, J. Knoll / Cooperative mechanism

not be regarded as a contradiction of the consideration in subsect. 2.1 where we used the cascade to determine the interaction-contact among nucleons. We take this rather to be a first-order approximant for the a&M, N) assuming that any refined dynamical scheme like off-shell scattering processes may not alter them on the average. The spectra QN(Py), on the other hand, are very sensitively dependent on the underlying dynamics. Here we want to relax the constraints imposed by the on-shell conditions of the cascade. In view of the complexity of the process we choose the ultimate possibility of imposing no intermediate-energy constraints. The picture we adopt is that of the statistical limit: all states available to a particular cluster (M, N) by the conservation laws of baryon number, energy and momentum are accessible with equal probability, i.e. the phase,space limit. This is the picture which allows for an optimal pooling of energy. Thus, the spectra FL.,(&) are determined by the number of final states, given the fact that a pion with momentum pr is observed, divided by the total number of states accessible to the cluster (M, N). The required level counting demands specification of the volume V,, of the system. Lack of better insight leads us to take this proportional to the nucleon number of the cluster, i.e. V,, pc = M + N by means of a density parameter pC.This parameter was fitted for one system. We found pC= p0 = 0.17 fmm3 to be optimal and kept it constant throughout. As an extension of earlier models the formation of deuterons, tritons and other light nuclei up to mass 12 is included in the final channel, such that the nucleon number of all products add to M + N; charge conservation was treated only on the average. Details will be given in appendix A. In summary, the model involves two parameters which were kept constant: CT~~ to determine the distributions of clusters from the nuclear geometry, and pC which specifies the level counting within the phase-space approach. In addition, the total energy and momentum available to each cluster are considered to fluctuate due to Fermi motion. This is included by a folding procedure described in appendix A. In our model the pion production via formation and decay of the nucleon resonances (deltas) has not been considered. One reason is the obvious fact that at these energies the likelihood of the formation of deltas is quite small. Secondly, if formation of the deltas is considered, it would lead to a shoulder in the inclusive pion spectrum around E, = 130 MeV [refs. 23.24)] a fact not corroborated by experimental data 4). 3. Discussion and results What is the physics implemented in our model? Assuming that the violent interactions among the basic constituents, the nucleons, are the key mechanism mediating the transport of energy and momentum, we consider the inclusive cross sections to arise from the incoherent contributions of all possible clusters. This assures a model that smoothly interpolates between the single-collision picture and the limit of global interaction-contact as in the fireball model. With regard to the

R. Shyam, J. Knoll j Cooperariw

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611

final channels of the reaction, we allow, besides the production of the pion, also the formation of composite nuclei, in principle up to the fused nuclear system. In this way our model interpolates up to the pionic fusion 25) limit. Two assumptions specify

the model:

(a) the distribution

of the cluster

sizes is essentially

determined

from the collision geometry, and we determine these distributions from our experience with the 3-dimensional cascade calculations; (b) in order to allow for a sharing of energy in a less constrained manner than in the cascade models we consider the statistical limit for the cluster dynamics, i.e. we assume for each cluster all possible final states (these include the formation of the pion and composite nuclei) with equal probability. In an earlier paper “) we showed that the NN single-collision mechanism is insufficient to explain the pion yields as one approaches closer and closer to the absolute threshold. Compared to earlier statements given by others ‘) we reached our conclusion through a careful description of the initial one-nucleon energymomentum distribution, which we took from the shell model. We further demon-

12

c+

‘*c-WI++x

E beam = 85 MeVlA

c

\\

\

\

\

\

lbL

\\

\

(a>

12oo \I

I

I 40

20

I 60

I 80

E,(MeV) Fig. la.

Invariant

cross section for the production

of T+ as a function of pion kinetic energy for several

pion angles. The colliding

system is ‘*C + “C and the beam energy 85 MeV/A. Solid curves represent the results of our calculations where we have assumed the formation of the fragments of masses up to 12 in the final channel calculations

alongside

the produced

pion. The dotted

curve represents

the results of our

(for the pion angle of 120”) when only nucleons exist along with the pion in the final channel.

612

R. Shyam, J. Knoll / Cooperative mechanism

strated that for an understanding of the shapes of the pion spectra, not only a cooperative pooling of energy is necessary, but also the accompanied formation of the composite nuclei. However, in that study no predictions for the absolute cross section could be given as we only required the final state to have a certain averaged mass of the associated fragments. In the present work we would like to demonstrate that by considering all possible fragmentation channels within the statistical limit one can calculate also the absolute cross sections with a reasonable choice of one parameter (p,). Moreover, with a constant value for this parameter the trends of the data with changes of both the bombarding energy as well as the nuclear masses are nicely reproduced. Let us see what the model can do. We begin by comparing our calculations with the experimental data for charged pions corresponding to beam energies around 85 MeV/A, where most of the recent experimental activities have been concentrated. In figs. la-ld we show the results of our calculations for the invariant cross section as a function of the pion kinetic energy for several pion angles. The projectile is 12C while the target nuclei span a wide range over the periodic table (‘Li to “*Pb), the beam energies being 85 MeV/A and 75 MeV/A. The experimental data have been taken from ref. ‘). It is clear that our calculations reproduce the shapes as well as the absolute yields of the data reasonably well for all the target nuclei. In order to emphasize the importance of the formation of fragments in the final channel, we also show in fig. la the result of our calculation when the final state comprises only nucleons and the pion (by dotted line) corresponding to the pion angle of 120”. In this case the

Ehom =85MeVlA

-

En (MeV) Fig.

1b. Same as fig. la for the ‘%I +‘Li system. Solid curves have the same meaning as in fig. la.

R Shyam, J. Knoll / Cooperative mechanism

Ebeom = 85 MeVlA _

En IMeW Fig. lc. Same as fig. la for the “C +*08Pb system.

En (MeVI Fig. Id. Same as fig. la for the 12C + 12C system but at a beam energy of 75 MeV/A.

613

614

R. Shyam, 1. Knoll / Cooperative

mechanism

absolute yield is down by nearly 3 orders of magnitude and also the slope is much steeper. This clearly shows that the formation of fragments in the final channel is very important. One can easily understand this result by noting that the formation of fragments is tantamount to the freezing of the kinetic degrees of freedom in the final channel. Consequently, in our statistical picture the available total energy will be shared less and hence more energy will become available to the outgoing pion. This would enhance the absolute yields and also the slopes of the spectra would become flatter. Note that uncertainties in the values of the formation cross sections uAB are quite unlikely to be of 2-3 orders of magnitude. Values of uAB calculated with the pure straight-line geometry and with the prescription presented in this paper differ only by factors of 3 to 4. Thus differences seen between full line and the dotted line in fig. la are to some extent due to the fragment formation in the final channel. Preliminary experimental studies have already been performed to check this prediction. The absolute yields of fragments of various charges have been measured 2”) recently in coincidence with pions in the collision of “C with “C at a beam energy of 85 MeVIA; relatively large yields of the light charge fragments were detected. More experimental studies along these lines would be very welcome. In fig. 2 we show the comparison of the calculated double-differential cross sections with the data for various target nuclei at a fixed pion angle of 90”. The target

E, IMeW Fig. 2. Double-differential cross section as a function of pion kinetic energy for a fixed pion angle of 90”. The projectile nucleus is 12C whereas target nuclei vary from ‘Li to “‘Pb. The beam energy is 85 MeV/A. Solid curves have the same meaning as in fig. la.

R Shyam, 1. Knoll / Cooperative

mechanism

615

0.1 0

20

40

60

80

100

120

En (MeVl Fig. 3. Angle-integrated cross sections for the production of T“ as a function of pion kinetic energy in the collision of “C with “C. The beam energies vary from 60 MeV/A to 84 MeV/A. The histograms represent the experimental data whereas the solid curves the results of our calculations.

dependence

of the double-differential

cross

section

is well reproduced

by our

calculations. Let us now focus our attention on extensive measurements of the neutral pion spectra performed by No11 er al. 4, and Heckwolf et al. ‘). In these studies the projectile nuclei involved are “C, “0 and 40Ar whereas the target nuclei range from C to U, and the beam energies vary between 44 MeV/A to 84 MeV/A. In fig. 3 we show the comparison of our calculations for the angle-integrated cross sections with the data for C +C collision at beam energies of 60, 74 and 84 MeV/A. We see that the calculations are in a reasonable agreement with the data for all three beam energies. We expect that some differences between experimental data and our calculations will show up in the extreme tails of the energy spectra. In this region one approaches the limit of the available phase-space. As the beam energy is lowered further the differences between our calculations and the data become more and more evident, because of the increasing importance of the one-body part of the interaction (mean-field effects). This is apparent from figs 4-6 where we compare our calculations with the data for energy- and angle-integrated cross sections for beam energies of 44 MeVIA and 35 MeVIA. The data has been taken from ref. 5, and ref. 6), respectively. Our calculations underestimate the data by factors of 1.5 to 5. However, it should be noted that the background has not been subtracted from

616

R Shyam, 1. Knoll / Cooperative e

*

”

*

t

I 40

20

t

mechanism

1

’

I!

“Ar + %a -mo+x

-

Ebeom= 44 MeVlA

-

I

L 60

I

I 80

,

E iMeW

Fig. 4. Same as in fig. 3 for the system 4oAr +40Ca at the beam energy of 44 MeV/A. The histogram and the solid curve have the same meaning as in fig. 3.

the data “) shown in fig. 6. This may account for some of the differences seen in this figure between the calculations and the experimental data particularly for larger pion kinetic energies. Nevertheless, collective processes are likely to be important at these lower energies. It would be interesting to see if our calculations combined with mean-field calculations in the one-nucleon model lo) provide a better description of the data. In any case, figs. 4-6 clearly show that even at lower bombarding energies the general trends of the experimental data are well reproduced by our model.

’

’

’

*

40Ar+4’fa >nn

1 g

-1 -0.8

I

-

’

no+

*

’

*

x

Ebeom= 4SMeVlA

’

1

-0.1

’

15 0 cos

”

0.4

1

I

0.8 1

(8)

Fig. 5. Energy-integrated cross sections for the production of w0 in the collision of 40Ar +%a at a beam energy of 44 MeV/A. The histogram and the carve have the same meaning as in fig. 3.

R. Shyam, 1. Knoll / Cooperative mechanism

617

1.2 1.0 0.8

0.6 0.4 0.2 0.0

2.0

1.0

,I E, (MeV)

Fig. 6. Same as in fig. 3 for the systems “‘N +“A1 and 14N +‘sNi at a beam energy of 35 MeV/A. histograms and the solid curves have the same meaning as in fig. 3.

200

300

The

400

P IMeW Fig. 7. The invariant cross section for the production of Q- as a function of pion momentum in the collision of “‘Ne with NaF at a beam energy of 183 MeV/A corresponding to several values of the angle of observation.

618

R. Shyam, _I. Knoll / Cooperative

In order to check the applicability it would be desirable to compare closer to the free NNP-threshold.

mechanism

of the model in the entire subthreshold

region

its predictions with the data for beam energies This has been attempted in fig. 7 where we

compare our calculations with the measurments of energy of 183 MeV/A. One notes that for this case reasonable agreement with the data, even in the tail pion momentum of 375 MeV/c. Note that already in by the phase-space model (without considering channel) was performed. However, in order to calculations required a pE which was twice the In the present work, on the other hand, we are p,,. This indicates that the fragment formation

Nagamiya et al. ‘) at a beam also our calculations are in a region which extends up to a ref. 3, an analysis of this data

the fragment formation in the final get an agreement with the data the normal nuclear matter density pO. able to fit the data with pC equal to somehow shows its effect even

at higher beam energies.

E (MeVIA) Fig. 8. The total T’ production cross section as a function of beam energy for the system ‘% + “C. The experimental data for beam energies blow 100 MeV/A have been taken from refs. 4*6) and those above this form ref. ‘). Where necessary the data have been translated to the “C + “C system by following an expression given in ref. 6).

In figs. 8 and 9 we have shown the total pion-production cross section as a function of the beam energy and the target mass, respectively. One notes from fig. 8 that the dependence of the cross section on the beam energy is rather well described by our calculations. The change of six orders of magnitudes which one sees in the experimental cross section in going from a beam energy of 35 MeV/A to 200 MeVIA is reproduced by our calculations without requiring any readjustment of the density

R. Shyam, J. Knoll / Cooperative I

-

1

111

mechanism I

12C+A --+x0+x

619

I

Ebeam

loo

200

400

A Fig. 9. The target-mass 74 MeVIA

60 MeV/A,

dependence of the total T’ production cross sections for the beam energies of points have been and 84 MeVIA (the projectile being ‘%). The experimental taken from ref. 4).

parameter. The experimental data show a dependence on the masses of the colliding partners in accordance with (ApAT)0.68, where A,, and AT are the masses of the projectile and target nuclei, respectively. Our calculations reproduce such a dependence for all three beam energies. In fig. 10 we show the invariant cross section as a contour plot in the plane of PJm, versus the parallel rapidity (Yll). In this plot the points of equal invariant cross sections are joined by contour lines. For “C + “C collisions the cross sections are forward-backward symmetric around 90” in the center of mass (c.m.) system (i.e. around the mean of the target and projectile rapidities equal to about 0.2), as it should be for identical colliding nuclei. On the other hand for the C +U system we notice that the center of the counters are shifted to a value of Yll smaller than 0.2. This is in accordance with the experimental data 4), although the shift observed in our calculations is somewhat smaller than that seen in the data (by 20-25%). It is remarkable, however, that our model shows a forward-backward assymetry without including the reabsorption and the Pauli blocking effects. In the NNSC model ‘) these are claimed to be the only sources which can introduce such an assymetry into the calculation. We feel that the effect of the Pauli principle should not be important in our calculations, because due to our multiple-collision picture the available phase-space is rather big ‘9*27).

R. Shyam, J. Knoll

620

/ Cooperative

mechanism

15 u ic 1.0

E

CL-+ 05

15

"! E 2

1.0

0.5

0.0

-1

0

1

Yll Fig. 10. Invariant rr” production cross section as a function of the parallel rapidity (Y,,) and the and ‘2C+23”U systems at a beam energy of perpendicular momentum (p,/m,c) for the “C+“C 84 MeV/A. The arrows labelled by T and P represent the positions of the target and projectile rapidities, respectively., whereas the dotted verical line the position of their mean. The additional arrow in the lower part of the curve shows the position of the center of the contours for the case of the “C +238U system. This is drawn to indicate the shift of the center of the contours for this case from the position of the mean of the target and projectile rapidities.

The discussions presented above, thus make it quite clear that our model is able to describe almost all the systematics of the data accumulated so far on subthreshold pion production. The remarkable fact is that no readjustments of the density parameter are needed while changing from one beam one set of collision partners to another.

energy

to another

or from

4. Conclusions In this paper we have presented a model for understanding the production mechanism of pions at subthreshold beam energies. This is based on the multiple collisions between several of the projectile and the target nucleons. Our multiplecollision dynamics, however, is different from that incorporated in the intranuclear cascade (INC) model. In the latter the collision between two nuclei is visualised as a sequence of classical binary on-shell nucleon-nucleon collisions. However, at these lower beam energies such a model would be no better than the NNSC model since due to the on-shell nature of the collisions only the first nucleon-nucleon collisions would be energetic enough to produce a pion. In our model, on the other hand, we take into account the quantum off-shell scattering between the nucleons. Thus a genuine cooperation among them is invoked in pooling their energies to

R. Shyam, J. Knoll / Cooperative

mechanism

621

produce a pion. The number of nucleons participating in the collision process has been estimated from a picture which in a way approaches the 3-dimensional cascade limit. The model involves a density parameter which we held fixed to a value equal to the normal nuclear matter density p. ( = 0.17 fmm3) for all the cases studied. The calculations performed within the model provide a good overall description of the data over a wide range of beam energies and colliding nuclei. The dependence of the pion-production cross section (which changes by 6 orders of magnitudes between the beam energies 35 MeV/A to 200 MeV/A) has been reproduced without requiring any readjustment of the density parameter. The target-mass dependence of the production cross section is also well described by our calculations. At lower beam energies the cross sections calculated by our model are somewhat smaller than the experimental data, particularly in the extreme tail region of the spectra. This is due to the fact that at lower beam energies mean-field effects are expected to manifest themselves rather strongly, something we have ignored. The present study suggests that cooperative action of several nucleons in the pion production at subthreshold beam energies is indeed very important. In the model presented here this enters through multiple nucleon-nucleon collisions and the fragment formation in the final channel. The latter should reflect itself in a correlation between the produced pion and the mass distribution of the associated nuclear fragments. More exclusive measurements of the pions in coincidence with the associated fragments would be interesting in this context. Useful conversations with P. Braun-Munzinger, J. Cugnon, E. Grosse, Jakobsson, U. Mosel and B. Schiirmann are gratefully acknowledged.

B.

Appendix A

The number of states per c.m. energy interval d EC.,, accessible to a microcanonical system enclosed in a volume V and subjected to the conservation laws of energy, momentum and baryon number (B) is given by h(E,.,., B) d&.,. =

C

Ul...fK)

(V/(2~)3)K-‘JK(E,.,.;

ml * * . mK) G.,.

Wfi

. * *_&I. (A.1)

Here {f, * * -fK} denote the possible composition of the system into K different fragments with masses m, baryon number bi and degeneracy Ai. The phase-space integrals JK determine the accessible momentum space volume, JK(Ec.m.;

m, .*-m,)=

d’p, . *. d3PK 8 (T &i-EC_-.} S’(:pi)

(A.2)

Ef= mf+pf.

(4.3)

with

R. Shyam, J. Knoll

622

The discrete

part of the spectrum

/ Cooperative

is governed

mechanism

by (A.4)

where B is the total baryon number of the system. In general, (A.4) can be supplemented by charge conservation as well. We shall, however, ignore it assuming it to be fulfilled on the average. The non-invariant phase-space integrals are undesirable for computational reasons. On the other hand, Lorentz-invariant phase-space integrals, apart from being suitable for calculations of the cross sections when one does not want to stick to one particularly chosen frame of reference, also involve some very handy recurrence relations 2”). These integrals are given by

~K(.%,.; ~1.

**

mK)=

W3p,lE,).. . (d3PK I&K)6

(; &i-E) a3(:Pi-P) 9

(A.3 where Ef.,, = E2-p2,

(‘4.6)

with E and P being the total energy and the total momentum of the system. Integrals (A.3) and (AS) are approximately related to each other via the relation 29) JK(E,.,.;~,...~K)=IK(E,.,.;~,...~K)

ir 6,

(A-7)

i=*

where the Ii are the appropriate non-relativistic limit Ei = mi.

means

of the energies

of the fragments;

in the

The study of pion production at subthreshold energies allows the following approximations: (i) The number of pions per baryon is small. This limits the final states to having zero or one pion at the most. Furthermore, the phase-space with one pion is much smaller than that without a pion. (ii) With the c.m. energy per baryon below 100 MeV we can use the non-relativistic limit of the phase-space integrals for all those compositions which involve no pion.

In the non-relativistic limit invariant phase-space integrals: IN,“(E,,)=(

(j,

a closed-form

mi)lii/T

X( EC,,.-:

expression

can be derived

mi) ((2~)3cK-“‘21(r(3(K

mi)‘3Km”‘2.

The partial probability to observe a pion at momentum given by the number of states of the nuclear complement

for the

-1)/z))

(A.81 p,,

with energy E, is then given the fact that this

R. Shyam, J. Knoll / Cooperative mechanism

623

pion is observed relative to the total number of states: &XpJ(d3p,le,)

= m,( Vl(2~)3)(d3p,le,)

B)/(n(dKF,

x(n(J(E-ET)*-(P-p_)*,

B)+r.t.), (A.9)

where n includes all compositions with no pion and r.t. stands for the neglected phase-space part with one pion. For a particular cluster (M, N) with baryon number B ( = M + N) the total energy E and momentum P are affected by the Fermi motion of the nucleons. Relations between E and P and total Fermi momenta (PM and I’,,,) of the group of M and N nucleons out of the nuclei A and B are given by E=ME,+(P,lM,)PM,+NM,, PII =

MPA

+ ( EA/

MA)&

+ PN,,

,

(A.lO)

p~=KtL+pNI,

written in the rest frame of the target B. Here EA, PA,hfAand MB denote the energy per nucleon, momentum per nucleon and masses per nucleon of the projectile A and the target B. Note that (A.lO) assumes the nucleons to be shell-model particles distributed in fixed orbits with binding energies BA and I& such that kfA=mO-BA,

M,=mo-Bg,

E$,=M:-PA,

(A.1 1)

where m. is the nucleon rest mass. The spectral distribution required in (2.1) is then given by a folding over all possible intrinsic momenta:

hdP,)

=

d’PAd3P,w~(f’,wh(f’,v)4ikr).

(A.12)

Here wM(PM) and wN(PN) denote the distributions of the momenta PM and PN in the respective rest frames of the two nuclei. We take these as gaussians with widths given by the Goldhaber rule ‘O): a:M=(M(A-M)/(A-l))fP~~,

(A.13)

CT;,=

(A.14)

(NW-N)I(A-I))fPjB,

where P,-and P/, are the Fermi momenta of a single nucleon in the nuclei A and B, respectively.

624

R Shyam, 1. Knoll / Cooperative

mechanism

References I) W. Bencnson et a/., Phys. Rev. Lett. 43 (1979) 683 2) T. Johansson et al., Phys. Rev. Lett. 48 (1982) 732; V. Bernard er at!, University of Lund preprint 8301 (1983) 3) S. Nagamiya et al., Phys. Rev. Lett. 48 (1982) 1780 4) H. NOB et aL, to be published; E. Grosse, Proc. Sixth High Energy Heavy Jon Study and Second Workshop on anomalons. Berkeley, June 28-July 1, 1983, p. 225 5) H. Heckwolf et aL, Z. Phys. A315 (1984) 343 6) P. Braun-Munzinger et aL, Phys. Rev. Lett. 52 (1984) 255 7) B. Jakobsson, Phys. Scripta TS (1983) 207 8) R. Shyam and J. Knoll, Phys. Len. MB (1984) 221 9) W.G. McMillan and E. Teller, Phys. Rev. 72 (1947) 1 IO) M. Tohyama, R. Kaps, D. Masak and U. Mosel, Phys. Lett. 1368 (1984) 226 II) J. Aichelin and G.F. Bertsch, preprint (1983) 12) D. Vasak, B. Miiller and W. Greiner, Phys. Scripta 22 (1980) 25; D. Vasak, H. Stocker, B. Miiller and W. Greiner, Phys. Lett. A360 (1980) 243 13) J. Knoll, Phys. Rev. C20 (1979) 779 14) J. Cugnon, J. Knoll and J. Randrup, Nucl. Phys. A360 (1981) 444 15) J. Gosset et aL, Phys. Rev. Cl6 (1977) 629 16) A.Z. Mekjian, Nucl. Phys. A312 (1978) 491; Phys. Rev. Cl7 (1978) 1051; S. Dasgupta and A.Z. Mekjian, Phys. Reports 72 (1981) 131 17) J. Knoll, Nucl. Phys. A343 (1980) 511 18) J. Hiifner and J. Knoll, Nucl. Phys. A290 (1977) 460 19) J. Cugnon, T. Mizutani and J. Vandermeulen, Nucl. Phys. A352 (1981) 505 20) J. Knoll and J. Randrup, Nucl. Phys. A324 (1979) 445 21) W. Myers, Nucl. Phys. A296 (1979) 177 22) S. Bohrmann, Phys. Lett. 107B (1981) 269 23) H.W. Bat-z, B. Lukas, J. Zimany, G. Fai and B. Jakobsson, Z. Phys. A302 (1981) 73 24) S. Bohrmann and J. Knoll, Nucl. Phys. A356 (1981) 498 25) K. Khngenbeck, M. Dillig and M.G. Huber, Phys. Rev. Lett. 47 (1981) 1654 26) M. Balore et al., Proc. Int. Workshop on gross properties of nuclei and nuclear excitations XII, Hirschegg, Austria (1984) p. 187 27) A.H. Blin, S. Bohrmann and J. Knoll, Z. Phys. A306 (1982) 177 28) P.P. Shrivastava and G. Sudarshan, Phys. Rev. 110 (1958) 765 29) R. Hagedom, Relativistic kinematics (Benjamin, New York, 1963) 30) A.S. Goldhaber, Phys. Lett. 53B (1974) 306