Dynamical and statistical aspects of intermediate energy nucleus-nucleus collisions

, in the reaction plane as a function of the rapidity, y = : £n[(E+p )/(E-p )), of the emitted particles, where E and p denote the total energy and L Z Z the momentum parallel to the beam axis. At the energy of E/A-50 Me~, fast particles (y>O) are preferentially emitted to negative values of

: negative deflection angles dominate due to the attractive nature of the mean field. At h~gher beam energies, E/A~150 MeV, n u c l e a r compression effects dominate and fast particles are preferentially emitted to positive values of

, i.e. to positive deflection angles. At present, there is no direct experimental confirmation about the occurrence of such a change of sign. Support for the model can be derived from a rather large number of successful comparisons between e x p e r i m e n t a l and theoretical transverse momentum distributions, see StOcker and Greiner (1986) for a recent review. However, the experiments performed at higher energies did not determine the absolute slgn of the transverse momentum distributions.

Nb+Nb, 150

I

I

b=3fm i

i

I00

50 -50

loll

"E/A =150MeV

E/A= 4 0 0 M e V

50

1

I

' E/A= 2 5 0 MeV

• E/A= 6 5 0 MeV

• •

E/A= 1.05 GeV

1

•

4

0

°

!

0 -I00 -150 -I

I -0.5

I 0

•

Ooe •

>~ -JO0 " E / A = 5 0 MeV ~;~ I00

OO°O

II•l

"7...L.

0

[

0.5

I

-0.5

½

t

0

0.15

o

";;; •

,

0

O15

/E+pz \

Fig. 17. Predicted d e p e n d e n c e of the mean transverse momentum,

, on the rapidity, y, of the emitted particles for Nb~Nb collisions at b=3 fm and for v a r i o u s i n c i d e n t energies. (Molitoris and StOoker, 1985)

4.3. Lisht particle correlations at larse relative momenta Additional information about the dynamical and geometrical aspects of intermediate energy nucleus-nucleus collisions can be obtained from detailed investigations of light particle correlations at large relative momenta for which final state interactions between the detected particles are expected to be negligible. Two-particle correlations at large relative momenta are sensitive to the importance of the phase space constraints for small ensembles due to conservation laws, collective flow effects and possible shadowing effects (Lynch and coworkers, 1982; Tsang and co-workers, 1984c). At sufficiently high energies, such correlations were used to evaluate the relative contributions of direct knock-out and multiple scattering processes; by investigating the dependence of these processes on the size of the target nucleus estimates of the nucleon mean path in nuclei were obtained (Tanlhata and co-workers, 1980, 1981). For intermediate energy nucleus-nucleus collisions, light particle correlations at large relative momenta were investigated by Lynch and co-workers (1982), Tsang and co-workers (1984c), Krlstiansson and co-workers (1985), Ardouin and co-workers (1985), Fox and co-workers (1986), and Chitwood and co-workers (1986a).

58

Claus-Konrad Gelbke and David H. Boal

S i n g l e - and two-partlcle inclusive cross sections exhibit rather similar dependences on energy and scattering angle of the emitted particles (Lynch and co-workers, 1982; F i e l d s and coworkers, 1 9 8 6 a ) . In o r d e r to e l i m i n a t e the strong d e p e n d e n c e of the single p a r t i c l e distributions on angle and energy of the emitted particles and to be sensitive to the dynamical aspects of the reaction, two-particle coincidence data are often presented in terms of the twoparticle correlation function C(AE,,AEz,e,,e=,$) which is defined as:

C(&E,,AE=,B,,e=,$)

$ dE, I dE= d~°*2 (E,,E=,e,,e=,¢) A E a A = + dE,dfl,dE=dfl=

&El

=

~ = _

I dE, .--.~-~.-(E,,e,) &E, dE,dfi,

,

(31)

e=)

• I dE= ~ U ~ ( E = dE=d~= aE=

'

where 8, and 8= denote the polar angles of particles I and 2, # is the d i f f e r e n c e of the azimuthal angles with respect to the beam axis, and AE, and AE= denote the intervals over which the cross sections are integrated. When singles and c o i n c i d e n c e cross sections are m e a s u r e d s i m u l t a n e o u s l y , the use of correlation functions has the additional advantage that systematic errors are cancelled which could arise from uncertainties in the energy, s c a t t e r i n g angle, or solid angle calibrations of the d e t e c t o r s . F o r the c a s e of u n c o r r e l a t e d emission, C(AE,,&E=,e~,8=,@)-constant, the two-particle cross sections are proportional to the product of the corresponding single particle cross sections. I

'

"

O"-A(ISO,

I

'

"

I

xy),

'

I

'

'

!

+

--

'

'

i

ELAB/A=25MeV,

O.I- x=p - Y=P

•

I

.

+.

'

I

'

•

T

8x = 4 0 ° ,

JI- x=p -I- y=d

o

.

i

,

JI- x=p -I- y=t

.

.

i

.

,

•

.

I

.

.

i

Ex,y=36-120Me~-

.+' ~

-I- x=d "1- y=d

,.

-t- x=crv "t- y=t

. , , : : ,I, : ; , : : :I, ; ; : ; : :I: x-" ' '"'x= ' ""x=d P P y=d - . " y=t ,J-y=d f - "

-

; ., : x=d y='

/

,

,, ,

~-

+-d,' +:J+ + +

I , , I

9o

.

+.,

b) A= mc °

I

8Y = 7 0 ° ,

+

b~ , . ,I, -" O"x=' .... '" 5 I P -..5.y=p -

l.,

.

moo

I , , I , , I

9o

I , , I . , I

,coo

so

moo

I * , I , , I

90

moo

I , , I , , I

90

,~

4' (decj) Fig. 18. Azimuthal correlations measured for *'O induced reactions on *'VAu (part a) and *=C (part b) at E/A-25 MeV. The c o i n c i d e n t light particles are detected at the p o l a r angles of 8,=40 ° and 8=-70 ° with respect to the beam axis. The particle energies were i n t e g r a t e d over the energy interval of A E , = A E = - 3 6 - 1 2 0 MeV. (Tsang and co-workers, 19840) Correlations measured for *=O induced reactions on *e~Au and *=C at E/A-25 MeV were measured by Tsang and co-workers (1984c) and Chltwood and co-workers (1986a). As an example, Fig. 18 shows the azimuthal c o r r e l a t i o n s between different light Particles detected at the polar angles of e,-40 ° and e,.70 °. The energy intervals AE,-AEs-36-120 MeV were chosen to e l i m i n a t e particles e m i t t e d at energies close to the exit channel Coulomb barrier which could have significant oontrlbutions from the later, more equilibrated stages of the reaction. As could already be e x p e c t e d from Fig. 13, the c o r r e l a t i o n functions have m a x i m u m values if the c o i n c i d e n t partloles are emitted in a plane which contains the beam axis ($-0 ° and 180o). For r e a c t i o n s induced on * ~ A u (part a), the correlations are nearly left-rlght symmetric with respect to the beam axis and exhibit a p r o n o u n c e d m i n i m u m at ¢-90 ° w h i c h becomes more p r o n o u n c e d w i t h

Nucleus - Nucleus Collisions

59

i n c r e a s i n g m a s s of the e m i t t e d particles. For reactions induced on *=C (part b), emission of two particles to opposite sides of the beam axis is more likely than emission to the same side of the b e a m axis, i.e. the c o r r e l a t i o n f u n c t i o n s have a m a x i m u m at ¢-180 °. This enhanced emission to o p p o s i t e sides of the b e a m axis becomes, again, m o r e p r o n o u n c e d for h e a v i e r particles. The qualitative features of the two-particle correlations for the 1 6 0 + * g V A u r e a c t i o n can be u n d e r s t o o d in terms of an o r d e r e d t r a n s v e r s e m o t i o n in the entrance channel reaction plane which is superimposed onto the random motion of the individual light particles. The s o l i d and d a s h e d curves in the Fig. 18a i l l u s t r a t e this effect. The dashed curves show the results of calculations in which the ordered motion was parametrized in terms of a rotating ideal gas (see Eqs. 27, 28); the solid curves show the results of calculations in which the ordered motion was parametrized in terms of two sideways deflected sources (Tsang and co-workers, 1984c). Momentum and energy conservation effects were neglected in these calculations. The calculations describe the q u a l i t a t i v e f e a t u r e s of the o b s e r v e d c o r r e l a t i o n s r a t h e r well. T h e y r e p r o d u c e the c h a r a c t e r i s t i c " V " - s h a p e of the azimuthal correlations and the experimental observation that the minimum at ¢-90 ° becomes more pronounced with increasing mass of the emitted particles. As w a s a l r e a d y d i s c u s s e d in c o n n e c t i o n w i t h Fig. 13, the m a s s d e p e n d e n c e of the a z i m u t h a l correlations can be interpreted in terms of the d e c r e a s i n g i m p o r t a n c e of thermal v e l o c i t y components for heavier particles which enhances the effects due to collective motion.

.41

t97Au (mO, xlxz),

E/A=25MeV, '

,

,

,

,

.

,

.

.

,

,

'

I

'

'

t

'

~

I

81=40°, '

'

"

"

gS=O° I

'

'

I

'

'

I

'

'

.

I .3 ~

,

~

~

- " .2 J~

-

,°

i

b

•N--.4

XI= p

Xl=p X2= t

xl=P x2=d

x2=p

,

,

t

,

I

,

,

I

=

,

=

=

i

,

,

i

,

,

i

,

'

'

I

"

I

'

'

I

'

'

'

'

I

'

'

I

'

'

i

'

"

::::',::',::

i

.5 oo .2

.,

,TI,

, , , , . -

9 0

0

9 0

xl--t , x2=t -90

0 90 02 (deg)

-90

0

90

Fig. 19. In-plane correlations between coincident light p a r t i c l e s e m i t t e d in a60 induced reactions on ~9~Au at E/A-25 MeV. Particle I is emitted at e~=40 °, particle 2 is detected in a c o - p l a n a r geometry. Positive values of 02 correspond to e m i s s i o n to o p p o s i t e sides of the b e a m axis; n e g a t i v e values correspond to emission to the same side of the beam axis. (Chitwood and co-workers, 1986a) I n - p l a n e c o r r e l a t i o n s m e a s u r e d for the a60+t97Au reaction at E/A=25 MeV are shown in Fig. 19 (Chitwood and co-workers, 1986a). The correlations are shown as a function of the polar angle, e=, of p a r t i c l e 2 w h i c h is detected in coincidence with particle I emitted at the scattering angle et-40 °. The two particles are detected in a plane which contains the beam axis. P o s i t i v e angles ®2 c o r r e s p o n d to e m i s s i o n of the two p a r t i c l e s to opposite sides of the beam axis, negative angles e 2 correspond to particles emitted to the same side of the b e a m axis. The Inplane correlations exhibit a pronounced minimum at small angles. The solid curves in the figure

PPHP-E

60

Claus-Konrad Gelbke and Dafid H. Boal

show correlations predicted by the parametrlzation of gqs. 27 and 28. The calculatlons predict a minimum of the correlation function at small angles in qualitative agreement wlth the experimental in-plane correlations. The occurrence of such a minimum at forward angles ls characteristic of systems for which the emission of particles is enhanced in a plane which contains the beam axls (Chltwood and co-workers, 1986a). However, the calculations do not reproduce the depth of the minimum of the in-plane correlations at forward angles. It may be possible that additional suppression of the correlation function at small angles is caused by I m p a c t parameter averaging effects. Low muitipllclty peripheral reactions which do not contribute to the coincidence cross sections could make increasing contributions to the single particle Incluslve cross sections at small angles resulting in a suppression of the correlation function. To address such questions, multiplicity selected two-partlcle correlation data are needed. Such data are not yet available. Particle correlations are influenced by phase space constraints imposed by conservation laws (Knoll, 1979, Lynch and co-workers, 19821Tsang and co-workers, 1984c; Chltwood and co-workers, 1986a). This influence can dominate the correlations measured for small systems, such as the ~60+12C system. The solid curves shown in Flg. 18b illustrate this effect. They show the results of schematic calculations with a moving source parametrlzatlon incorporating the effects of energy and momentum conservation (Lynch and co-workers, 19821 Tsang and co-workers, 1984c). In these calculations, the two particles were assumed to be emitted sequentially. For the emission of the first particle, a Maxwellian velocity distribution of temperature T,-7.3 MeV and mean velocity v0~- 0.13c, directed parallel to the beam axis, was assumed. (These parameters correspond to the compound nucleus velocity and temperature.) For the emission of ~he second particle, a Maxwelllan velocity dlstribution of temperature T 2 and mean velocity v02= IA0vo,-Alvl)/(A0-Al) was assumed where v~ and A~ denote the velocity and mass number of the first emitted particle, and Ao-28 is the mass number of the compound nucleus. Because of e ~ e r g y conservation, the temperature o~ the second source ~as reduced to the value of: T~=8e2/(Ao-A I) MeV; where el-¢2-m0(~1-~ol)~AoA1/2(Ao-Al), eI-AoT~/(8 MeV), and m o denotes the nucleon mass. The oalulatlons are symmetrlzed with respect to the sequence of emission of the two particles. They reproduce the enhanced emission of coincident particles to opposite sides of the beam axls and the increasing importance of thls effect for heavier particles, indicating the dominance of phase space constraints due to momentum and (to a lesser extent) energy conservation (Tsang and co-workers, 1984c, Chltwood and co-workers, 1986a). The detailed shapes of the azimuthal distributions are not reproduced by the s c h e m a t i c calculations shown in Flg. 18. Close Inspection of Flg. 18a reveals a small enhancement of the coincidence cross section at ¢-180 °, corresponding to the preferential emission of coincident protons to opposite sides of the beam axis. Coincident deuterons and tritons, on the other hand, are preferentially emitted to the same side of the beam axis. It is possible that these in-plane asymmetries are the result of the competing effects of momentum conservation and the refractive or absorptive ("shadowing") interactions of the outgoing particles wlth the heavy r e a c t i o n r e s i d u e ( T s a n g and c o - w o r k e r s , 1984c). The detailed shapes of two-particle correlations will also depend on details of the reaction dynamics. Different two-partlcle correlations may arise for systems which disintegrate instantaneously or via a sequence of statistical decays. The comparison of two-particle correlation functions with schematic parametrlzations can only serve to elucidate certain general features of the two-particle distributions which should ultimately be reproduced by more complete models of noncompound light partlcle emission. We expect that the qualitative features of the correlations can also be reproduced by other models which incorporate the superposltlon of collective and random velocity components and the phase space constraints caused by energy and momentum conservation. Two-partlcle correlations measured for 32S induced reactions on Ag at E/A-22.5 MeV (Fields and co-workers, 1986a) and for ~°Ar induced reactions on X'TAu at E/A-60 MeV (Ardouln and coworkers, 1985) exhibit qualltatlveiy slmllar, "V"-shaped azimuthal correlations as those shown in Fig. 18a. For the ~°Ar+~97Au reactlons, the In-plane correlations exhibit minima at forward angles similar to the those shown in Fig. 19. The correlations measured for the ~°Ar+t''Au reaction at E/A-60 MeV are, however, significantly less pronounced indicating a higher degree of randomization (thermallzatlon) of the particle velocities for thls reaction. Higher degrees of randomization may be expected because of the increasing importance of individual nucleonnucleon colllslons at higher energies. In addition, the larger number of participant nucleons should reduce the importance of particle escape from the surface of the reaction volume at the early stages of the reaction leading to increased thermailzatlon of the partlcle velocities. Unfortunately, existing data are insufflclent to delineate the dependence of two-partlcle correlations on projectile energy or mass. At present, the relative importance of random and collective velocity components is not well understood.

Nucleus

- Nucleus

61

Collisions

Evidence for the importance of single step knock-out reactions in high energy p r o t o n - n u c l e u s and n u c l e u s - n u c l e u s collisions was reported by Tanihata and co-workers (1980, 1981). However, attempts to i s o l a t e a c l e a n s i g n a t u r e for s i n g l e step k n o c k - o u t r e a c t i o n s have not been s u c c e s s f u l at i n t e r m e d i a t e e n e r g i e s (Fox and co-workers, 1986; Kristiansson and co-workers, 1985; Lynch and co-workers, 1982). Most likely, this f a i l u r e can be a t t r i b u t e d to the Fermi m o t i o n of the nucleons which smears out kinetic signatures of nucleon-nucleon collisions. Due to the increasing importance of r e s c a t t e r i n g effects for h e a v i e r nuclei ( T a n i h a t a and cow o r k e r s , 1980, 1981), s e a r c h e s for direct knock-out processes were performed for relatively small nuclear systems, such as '2C+'2C (Kristiansson and co-workers, 1985; Fox and c o - w o r k e r s , 1986). For these systems, the phase space constraints imposed by momentum conservation can lead to correlations which are difficult to distinguish from those due to knock-out reactions in the p r e s e n c e of F e r m i m o t i o n a n d the n u c l e a r m e a n field (Lynch and c o - w o r k e r s , 1982). At present, the identification of knock-out r e a c t i o n s for i n t e r m e d i a t e energy n u c l e o n - n u c l e o n c o l l i s i o n s is h a m p e r e d by the lack of t h e o r e t i c a l investigations which could delineate the differences expected for the various competing reaction mechanisms.

5. EMISSION OF PIONS AND HIGH ENERGY PHOTONS

5.1. Emission of pions P i o n s have the rest masses of m(~°)=135 MeV/c 2 and m(~+)=m(w-)=139.6 MeV/c 2. In free nucleonnucleon collisions, the production of pions is, therefore, only p o s s i b l e for beam e n e r g i e s l a r g e r than about 280 MeV. In nucleus-nucleus collisions, pions can be created at projectile energies significantly lower than 280 MeV per nucleon. In fact, the first pions ever created in the laboratory were produced in ~He induced reactions reactions on *9~Au at E/A=95 MeV (Gardner and Lattes, 1948a, 1948b). The production of pions at projectile energies per nucleon below the free nucleon-nucleon threshold is possible because of the velocities of nucleons inside nuclei (due to Fermi or thermal motion) and b e c a u s e of the p o s s i b l e p o o l i n g of energy by s e v e r a l n u c l e o n s . For intermediate energy nucleus-nucleus collisions, the production of pions requires the transformation of a m a j o r part of the total a v a i l a b l e energy into a single d e g r e e of freedom. One may, therefore, hope to obtain information about possible cooperative aspects of the reaction mechanism by studying the production of pions at energies significantly b e l o w the free nucleon-nucleon threshold. Over the last few years a considerable effort has been devoted to the m e a s u r e m e n t of pion p r o d u c t i o n cross s e c t i o n s at these " s u b - t h r e s h o l d " e n e r g i e s ( B e n e n s o n and co-workers, 1979; Chiavassa and co-workers, 1984; Grosse, 1985; Heckwolf and coworkers, 1984; Johansson and co-workers, 1982; Erebs and c o - w o r k e r s , 1986; N a g a m i y a and cow o r k e r s , 1981, 1982; Noll and c o - w o r k e r s , 1984; Stachel and co-workers, 1986; Young and coworkers, 1986).

t

,

i

i

i

i

i

I0 ~

==

I0::'

~,i,#"~"

I0 I

..".-" "

o,i =.¢=

i00

IO-=

3 IO-Z

a/,'

"

iII i0-~

'Ill

20

~

I

I

I

I

40 60 80 Ela b (MeV/nucleon)

I

I00

Fig. 20. I n t e g r a t e d cross s e c t i o n s for the e m i s s i o n of neutral pions in intermediate energy nucleus-nucleus c o l l i s i o n s . The c a l c u l a t i o n s are explained in the text. (Stachel and co-workers, 1986)

62

Claus-Konrad Gelbke and David H. Boal

Figure 20 shows the energy dependence of the integrated ~°-productlon cross sections (Heckwolf and co-workers, 19841Noll and co-workers, 19841Stachel and co-workers, 1986). To facilitate the comparison of measurements for different projectile target combinations, the cross sections were divided by (A AL)- ~. The measured cross sections are significantly larger than predicted by slmple nucleon-nucleon scattering mechanisms (Shyam and Knoll, 1984a; Guet and Prakash, 1984). Thls is illustrated by the dotted ilne in Fig. 20 which shows the cross sections predicted by the single nucleon-nucleon hard scattering model of Guet and Prakash (1984). Calculations based on the assumption of statistical emission from subsets of nucleons, on the other hand can reproduce the energy dependence of the measured cross sections rather well (Aichelin and Bertsch, 19841Aichelin, 19841Shyam and Knoll, 1984a, 1984b; Prakash and coworkers, 1986). For illustration, the solid line shows the results of thermal calculations by Prakash and co-workers (1986). In these calculations, the statistical emission of pions from a nuclear fireball is calculated from the Weisskopf formula (Alchelin and Bertsoh, 1984). The dashed llne shows the energy dependence of the plon cross sections predicted by the phase space model of Shyam and Knoll (1984a, 1984b), renormallzed by a factor of 0.2 (Shyam and Knoll, 1986). In these calculations the cross sections are calculated from the phase space available for "virtual clusters" of nucleons. These virtual clusters contain all those projectile and target nucleons which interact with each other. The distribution of virtual clusters was extrapolated from intra-nuclear cascade calculations performed at higher energies. Both calculations describe the overall energy dependence of the plon cross sections reasonably well. However, they fail to reproduce the cross sections at the lowest measured energy. Since the simple assumptions about the sizes and excitation energies of the decaying nuclear subsets are not expected to be realistic at lower energies, this failure Is not too surprlzlng. At projectile energies above E/A-60 MeV, the cross sections for the emission of pions can be rather well understood in terms of the Boltzmann-Uehling-Uhlenbeck theory (Aichelin, 19851 Bertsch and co-workers, 19841 Kruse and co-workers, 1985a). The inclusion of Pauli blocking and the mean nuclear field decrease the plon yields predicted in terms of this theory as compared to the intra-nuclear cascade model which overpredicts the plon production cross sections c o n s i d e r a b l y . For intermediate energy nucleus-nucleus collisions, the effects of plon reabsorptlon are associated wlth considerable theoretical uncertainties; In the calculations of Aichelln (1985) they were included in a phenomenological way. Experimental evidence for the importance of plon reabsorption was derived from the measured rise of the pion cross sections at backward angles (Stachel and co-workers, 1986). With the exception of energies close to the absolute threshold, the energy spectra of plons emitted at intermediate rapldities are qualitatively similar to those measured for noncompound light particles, indicating that plons and energetic light particles may be emitted In similar classes of reactions. For nucleus-nucleus collisions above E/A-IO0 MeV, the slope parameters which characterize the shapes of plon spectra are smaller than those characterizing proton energy spectra (Nagamiya and co-workers, 19821Krebs and co-workers, 1986). These differences have been interpreted as due to the interplay between statistical and collective velocity components during the explosive disintegration of highly excited nuclear systems (Siemens and Rasmussen, 1979), the sequential freeze-out of plonic and nucleonic degrees of f r e e d o m (Nagamlya, 1982) or due to the decay kinematics of produced a's (Brockmann and co-workers, 1984). Pions emitted with veloclti~s close to the projectile velocity exhibit a sharp peak in the cross section ratio, o(~-)/o(~ ), (Benenson and co-workers, 19791 Chiavassa and co-workers, 1984). Thls enhanced emission of negatively charged plons at velocities close to the projectile velocity Is qualitatively understood in terms of the distortions of the N o n wave functions In the Coulomb field of the projectile (Benenson and co-workers, 1979). If the plons are emitted fro~ a nuclear subsystem close to statistical equilibrium, the energy dependence of the ratio e(~ )/a(~ ) can serve as a useful probe of the nucleonic density of states (or temperature) at the early stages of the reaction (Bertsch, 1980b). At energies close to the absolute threshold, the kinetic energy spectra of the emitted pions exhibit little sensitivity to the projectile energy. Instead, the differential cross sections appear to depend primarily on the energy which is available in the center-of-mass system after the emlsslon of the plon (Stachel and co-workers, 1986). A satisfactory interpretation of the differential cross sectlons close to the klnematical limit is not yet available. Vasak and co-workers (1980) have polnted out, that "subthreshold" plons may also be emitted In a process analogous to bremsstrahlung if the two collldlng nuolel are slowed down sufflclently rapidly. Plon productlon cross sections and energy spectra measured for *zC+*2C oolllslons at E / A = 6 0 , 74, and 84 MeV c o u l d be rather well reproduced by model calculations using a phenomenologloal slowlng down functlon with a slngle deceleration parameter, A-1.8 fm (Vasak

Nucleus

- Nucleus

Collisions

63

and co-workers, 1980, 1984; Grosse, 1985). The dot-dashed lines in Fig. 20 show cross sections predicted by their model. The deceleration of two nuclei over such short distances must he a o c o m p a n l e d by the emission of high energy photons. The assumptions of the model can, therefore, be tested by measuring the differential cross sections of high energy photons emitted In nucleus-nucleus collisions.

5.2. Emission of hl~h energy photons High energy photons provide a unique probe to study the early stages of violent nucleus-nucleus collisions. This particular sensitivity is derived from the fact that reabsorption affects are negligible for electromagnetic radiation but significant for pions, nucleons and bound light particles. In addition, hlgh energy photons can provide information about the r e l a t i v e importance of coherent nucleus-nucleus bremsstrahlung, nucleon bremsstrahlung in the mean nuclear field, and bremsstrahlung from individual proton-neutron collisions (Ko and co-workers, 1985; Nifenecker and Bondorf, 1985; Bauer and co-workers, 1986a). The emission of high energy photons ( E ~ 5 0 MeV) in intermediate energy n u c l e u s - n u c l e u s collisions has been measured by Beard and co-workers (1985), Grosse and co-workers (1986), Kwato NJock and co-workers (1985), Alamanos and co-workers (1986), and Stevenson and co-workers (1986). Because of non-trlvlal experimental problems, rather large discrepancies exist between measurements performed with different detection techniques. Alamanos and co-workers (1986) investigated *~N induced reactions on Ni at E/A=35 MeV using Pb-glass Cerenkov detectors; they report angular distributions with pronounced maxima at ®i "" 90°' strongly reminiscent of a dipole radiation pattern. These results are in contrast W ~ h measurements of Stevenson and coworkers (1986) and Kwato Njock and co-workers (1986). Stevenson and c o - w o r k e r s (1986) investigated *~N induced reactions on C, Zn and Ph at incident energies of E/A=20, 30, and 40 MeN using an active CsI converter followed by a novel Cerenkov-range detector stack. Kwato Njock and co-workers (1986) investigated ~°Ar induced reactions on l'VAu at E/A=30 MeV using an active BaFp converter followed by two thin plastic scintillators and a 20 cm thick NaI detector. These authors report angular distributions which are slightly forward peaked in the laboratory rest frame, resembling emission from a thermal source moving with approximately half the beam velocity.

104 103

o" **c("~,WJ •

=

E/A

o o

102

101 m

:m 10 0

"..~~%

.o..v

~ 10_1 '0 b "0 10-2 10-8

i, ........ .T, .~.T., .'?E .°.O.i)~ 40

60

BO

100

120

E~ [MeV] Fig. 21. Photon energy spectra at 81=h=90° for 12C+*~N collisions at E/A=20, 30, and 40 M~e~ (solid points) and '2C+~2C collisions at E/A-84 MeV (open points). The solid lines are the results of microscopic calculations within the framework of the BUU theory. (Bauer and co-workers, 1986b)

Claus-Konrad Gelbkc and David H. Boal

64

Figure 21 shows the energy spectra of high energy photons measured at 8_ _-90 ° for ~N+*2C collisions at E/A-20, 30 and 40 MeV (Stevenson and co-workers, 1986) and f o r ~ C ÷ * 2 C collisions at E/A=84 MeV (Grosee and co-workers, 1986). The measured cross sections are significantly larger than predicted for coherent nucleus-nucleus bremsstrahlung (Bauer and co-workers, 1986a; Grosse and co-workers, 1986). The solid lines in Fig. 21 show the results of recent microscopic calculations with the Boltzmann-Uehling-Uhlenbeck theory (Bauer and co-workers, 1986b). In view of the experimental uncertainties, the agreement between theory and experimental data is reasonable. According to these calculations, high energy photons are primarily produced via proton-neutron bremsstrahlung at the very early stage of the collision; more than half of the cross section for the emission of high energy photons can be attributed to s i n g l e p-n scatterings. Several other investigations have also concluded that high energy photons are emitted prior to the attainment of full statistical equilibrium of the composite system (Grosse and co-workers, 1986; Ko and co-workers, 1985; Kwato Njock and co-workers, 1986; Nifenecker and Bondorf, 1985; Stevenson and co-workers, 1986). Future coincidence studies of high energy photon emission promise new and unique insight into the early stages of nucleus-nucleus collisions. At present, however, the discrepancies between different measurements need to be resolved.

6. EMISSION OF INTERMEDIATE MASS FRAGMENTS As the excitation energy per nucleon is increased, new mechanisms involving the emission of complex nuclei become important. Processes leading to the emission of intermediate mass f r a g m e n t s (Z>2) are of p a r t i c u l a r interest since they might provide key experimental information about the liquld-gas phase diagram of nuclear matter. More generally, microscopic d e s c r i p t i o n s of n u c l e a r fragmentation and the emission of complex nuclei present new theoretical challenges since they require theories which go beyond the calculation of one-body densities. In this section, we will examine the emission of intermediate mass fragments. We will review current experimental information, the physics involved with the l i q u l d - g a s transition of nuclear matter, and recent microscopic calculations of nuclear fragmentation. 6.1. Mass distributions The emission of intermediate mass fragments in processes different from binary fission was first observed in high and intermediate energy hadron-nucleus collisions and has b e e n associated with the most violent of these reactions (Poskanzer and co-workers, 1971; Westfall and co-workers, 1978; Wilkins and co-workers, 1979; Galdos and co-workers, 1979; Green and Korteling, 1980; Finn and co-workers, 1982; Hirsch and co-workers, 1984; Minich and co-workers, 1982). Figure 22 shows the mass distribution measured for proton induced reactions on xenon in the i n c i d e n t e n e r g y r a n g e of 80-350 GeV (Finn and co-workers, 1982). Except for some characteristic structures for lighter masses, A515, the mass distribution can be rather well described by a power law, OA~ A t with ~-2.64. Similar fragment distributions have since been observed for reactions over a large range of incident e n e r g i e s and p r o j e c t i l e t a r g e t combinations. The significance of the approximate power law dependence will be discussed later. More recently, the emission of intermediate mass fragments has been observed in nucleus-nucleus collisions over a large range of energies (Borderie and co-workers, 1984; Chitwood and coworkers, 1983; Fields and co-workers, 1984, 1986a; Frankel and Stevenson, 1981; Gutbrod and coworkers, 1982; Jacak and co-workers, 1983; Kwiatkowski and co-workers, 1986; Lynen and coworkers, 1982; McMahan and co-workers, 1985; Mittig and co-workers, 1985; Sobotka and coworkers, 1983, 1984; Trockel and co-workers, 1985; Warwick and co-workers, 1982, 1983). For intermediate energy nucleus-nucleus collisions, the emission of intermediate mass fragments already sets in during the early, non-equilibrated stages of the reaction (Fields and coworkers, 1984, 1986a; Kwiatkowskl and co-workers, 1986). As a typical example, Fig. 23 shows energy spectra for Li, Be, B, C, N and O nuclei emitted in S2S induced reactions on AE at E/A-22.5 MeV (Fields and co-workers, 1986a). The energy spectra exhibit qualitative features similar to those established for the emission of noncompound light particles (see Fig. 10): the slopes of the energy spectra become steeper with increasing emission angle, and the angular distributions are forward peaked in the center-of-mass frame of reference. The energy spectra are peaked at energies below the exit channel Coulomb barrier possibly indicating emission from h i g h l y d e f o r m e d systems. Fields and co-workers (1986a) analysed the energy spectra of intermediate mass fragments (Z-3-24) emitted in this reaction by adopting a t w o - s o u r c e parametrlzation. One source was assumed to correspond to the emission from the equilibrated composite nucleus formed in incomplete fusion reactions and the other (nonequilibrium) source was adjusted to provide a fit to the data (Fields and co-workers, 1986a). The fits, shown as the solid lines in Fig. 23, are superior to fits with a single moving s o u r c e . The fit

Nucleus

Nucleus Collisions

-

I

I0 o

I

I

I

65

I

I

I 25

I° 30

A-f2.64

iO'r

(n II0 e o u •

•

o•

iO s

I 5

104

I I0

I 15

I 20

Af Fig. 22. Mass distributions of fragments emitted in p+Xe collisions over the proton energy range of 80-350 GeV. (Finn and coworkers, 1982).

I-

10 4

I ....

~("s,x)

I'''l

....

I ....

E/A =

ss.5

I ....

I'"

Mev

10 3 x...

10 2

4oi.

101 10 0

.m 1 0 - 1 >. 10 3 ,~

102

,.~

10 0

101

10-1

~f

103 102 101

10 0

,11

II

...I

I

....

i I .i;.. "1 I I'1

r,x\\% r"

X-N

X-0

10-1 100 200 300 0

100 200 300

ENERGY (MeV) Fig. 23. Differential cross s e c t i o n s for Li, Be, B, C, N, and 0 nuclei e m i t t e d at 27.5 ° , 40 c, and 52.5 ° for '2S induced r e a c t i o n s on AE at E / A - 2 2 . 5 M e V . T h e s o l i d c u r v e s c o r r e s p o n d to fits with a t w o - s o u r c e p a r a m e t r i z a t i o n . (Fields and co-workers, 1986a) p a r a m e t e r s i n d i c a t e that the relative contribution from the fuslon-llke source increases wlth Inereaslng fragment mass, becomlng the dominant contribution for fragments heavier than oxygen. The velocity of the non-equlllbrlum source decreases with fragment mass, rendering the spectral

66

Claus-Konrad Gclbke and David H. Boal

decomposition Into two sources increasingly artificial for Z>13. Temperature parameters for the nonequlllbrium source are of the order of T-I0±2 MeV; they increase slightly as a function of fragment charge. Thls dependence differs from reactions with lighter projectiles, where the temperature parameters decrease systematically for heavier fragments (Fields and co-workers, 1984; Trockel and co-workers, 1985). The kinetic energy spectra of heavier fragments become i n c r e a s i n g l y sensitive to non-thermal velocity components due to Ferml-motion, Coulomb repulsion, or collective motion. As a consequence, the extracted temperature parameters do not necessarily reflect the temperature of the emitting system (Aichelin and HUfner, 1984; Alchelln and co-workers, 1984; Fields and co-workers, 1984; Hlrsch and co-workers, 1984; Ban-Hao and coworkers, 1985). Figure 24 shows the dependence of the fragment cross sections on fragment charge, Z, for the ==S+Ag reaction at E/A=22.5 MeV (Fields and co-workers, 1986a). The open points show cross sections extrapolated by integrating the fits over 4~ and all energies. The solid points represent cross sections, , obtained by averaging the experimental data over scattering angle. Both cross sections decrease smoothly with fragment charge and show no particular enhancement in the neighborhood of the projectile, Z=16, as would be expected at angles closer to the grazing angle. The elemental distributions fall off much more gradually with fragment

_

"1 .... I .... I .... I .... I .... Ag(SaS,X) E/A=22.5

O~_ ' ~

105

O

•

"

MeV

0"x

A 105 v

M

b

10 4

!

104~"

IIQ

103

,,I .... 0

5

[ .... tO

I .... 15

] .... 20

103

I,,, 25

30

Zx

Fig. 24. Integrated cross sections for intermediate mass fragments emitted In ==S induced reactions on Ag at E/A=22.5 MeV. The open points represent cross sections extrapolated from the fits shown in Flg. 23; the solid points represent the differential cross section, , averaged over the measured angular range. (Fields and co-workers, 1986a) charge than the yield distributions for proton induced reactions (see Fig. 22) or for *=Cinduced reactions (Chltwood and co-workers, 1983). For a system in chemical equilibrium, the relative fragment yields are closely related to the entropy per nucleon of the system. First attempts (Siemens and Kapusta, 1979) to extract the entropy per nucleon produced in heavy ion induced reactions employed the relative yields of protons and deuterons and the simple equilibrium relation, Eq. 10. Unexpectedly large entropies of S/A>4 were extracted. It was later realized, that this value was too large because of the neglect of decays of particle unstable states (StOcker and co-workers, 1983). Attempts to d e t e r m i n e the entropy from the relative abundances of intermediate mass fragments were undertaken by Jacak and co-workers (1983, 1984) whose results are shown in Fig. 25. The extracted entropy values do not exhibit a significant dependence on bombarding energy. Although the analysis incorporated effects from particle decays of low lying states, the extracted entropy values were found to depend on the range of fragment masses included in the analysis: they decrease when heavier fragments are included in the analysis. The assumption of chemical equilibrium does not provide a unique value of the entropy when compared to fragmentation data.

Nucleus

I

I

I

I

-

Nucleus Collisions

I111t

I

I

I

3.0

I1[111

I

C+Ag C+Au x Ne+Au

**++

u') 2,0

'

I

][111

• Ne+U A p÷Ag V p+U

+

+

1.0

67

+

HEAVY FRAGMENTS

I

I

Illll

I

J

I

I

Illil

I

I

i

I

I

I

5.0 4.(~ (I) 3.0

-

~'.~_.~,.,._p.

.,=

4

,-

2,0 I.C

3.0 i

I0

i

I

IIiiii

i

IOO

I

I

I11111

l

IOOO

I

SO00

Elob(MeV/Nucleon) Fig. 25. E n t r o p i e s e x t r a c t e d f r o m the r e l a t i v e f r a g m e n t y i e l d s m e a s u r e d for v a r i o u s projectile target combinations. The curves l a b e l l e d 1.0, 2.0, and 3.0 are e n t r o p i e s of an ideal Fermi gas of I, 2 and 3 times normal nuclear matter d e n s i t y with t e m p e r a t u r e s d e t e r m i n e d by the f i r e b a l l model. (Jacak and co-workers, 1984)

6.2. Coincidence measurements with intermediate mass fragments Most m o d e l s for the e m i s s i o n of intermediate mass fragments have only been compared to mass distributions. T h e y all have a c h i e v e d a c e r t a i n degree of s u c c e s s in these c o m p a r i s o n s . C r i t i c a l judgements or improvements of the various models can be expected only in the light of more restrictive coincidence measurements. Up to now, only few such i n v e s t i g a t i o n s have been performed. Coincidences between intermediate mass fragments and heavy reaction residues were i n v e s t i g a t e d by Fields and co-workers (1986a). Angular and velocity distributions of heavy reaction residues detected in coincidence with lithium and carbon nuclei emitted at 6~=27.5 ° are s h o w n in Fig. 26. T h e s e d i s t r i b u t i o n s represent probability distributions for the detection of target-like residues in coincidence with Li and C nuclei in the m o m e n t u m ranges P ( L I ) = 9 6 0 - 1 2 8 0 M e V / c , P(C)=1280-1600 and 1920-2250 MeV/c. The peak positions expected for complete fusion followed by binary decay are indicated by arrows. The average recoil velocities are lower and the maxima of the a n g u l a r d i s t r i b u t i o n s are located at angles larger than expected for such fuslon-fisslon processes indicating that the two d e t e c t e d nuclei do not carry away the entire p r o j e c t i l e m o m e n t u m . F i e l d s and c o - w o r k e r s (1986a) estimated that the probability for the emission of a heavy residue in coincidence with an intermediate mass fragment was of the order of unity: the a v e r a g e number of heavy recoil nuclei is approximately one, consistent with the existence of a single t a r g e t - l l k e r e s i d u e r e m a i n i n g after f r a g m e n t emission. A m o r e d e t a i l e d k i n e m a t i c analysis revealed that intermediate mass fragments were emitted in highly inelastic collisions. An amount of e n e r g y b e t w e e n 200 and 400 MeV was e s t i m a t e d to be d e p o s i t e d into i n t e r n a l excitations of the residual nucleus or emitted fragments. Considerable amounts of kinetic energy and linear momentum are carried away by light p a r t i c l e s w h i c h are e m i t t e d prior to e q u i l i b r a t i o n . F i g u r e 27 shows energy spectra of protons (upper part) and alpha particles (lower part) measured in coincidence w i t h l i t h i u m (left hand side)

68

Claus-Konrad Gelbke and David H. Boal

and carbon fragments (right hand side). Protons and alpha particles were detected at the polar angles of e -40 ° and e - 70 o and the azimuthal angles of @ -90 ° (open points) and ~ -180 ° (solid p o i n t s ) ; i n ~ e r m e d l a t e mass fragments were dete~ted at 8 - 2 7 . 5 ° and @x-O ~. For comparison, the solid curves show the respective inclusive energy spectra. Coincidence spectra measured for the coplanar geometry (solid points) are nearly identical In shape to the single particle inclusive spectra. Coincidence spectra measured for the out-of-plane geometry (open points) exhibit slightly steeper slopes. Thls difference in slope Is more pronounced for alpha particle spectra than for proton spectra. The differences in the temperature parameters which characterize the singles and coincidence spectra are, however, small. The largest measured difference between these parameters is of the order of 10-15%. Close similarities between single and two-partlcle inclusive energy spectra, measured in co-planar geometries, were also observed for ~ N induced reactions at E/A-tO-12 MeV (Bhowmlk and co-workers, 1981; 1982) and for ~2C and ~°Ar induced reactions at E/A=30 and 92 MeV, respectively (Hasselqulst and coworkers, 1985). The similarity of light particle spectra measured inclusively or in coincidence with intermediate mass fragments indicates that intermediate mass fragments and non-compound light particles are emitted in similar types of reactions.

I

I

"

"

-I

I

I

-

E/A=88.5 ~/eV, S1=27.5 °

Ag(~,X)

!2o

(a) X=Li , Pt=960-1280 MeV/c 0.01

.... ;. ..... 1 l,,

0.00

: ',....

0~

(b) X=C , Pz=1280-1600 MeV/c

lO

ill

I~O.O1

leeeleee°eeli%elle

.

•

-~

•

u'

l

~o.oo

g:

.

20 v

(c) X=C , PI=I980-2R40 MeV/c e

0.01

eeo

eoe%%t •

oe -

•

•

e,e

•

el

o.oo

.... 0

•

I. . . . . . . . 10

20

O z (deg)

,'--, 0

f" ....... 1

o

2

V z (cm/ns)

Fig. 26. Angular and velocity distributions for heavy recoil nuclei detected in coincidence with lithium and carbon fragments emitted at e,=27.5 ° for ~2S induced reaction on ~'~Au at E/A=22.5 MeV. The arrows show the values expected for binary reactions. (Fields and co-workers, 1986a) The azimuthal correlations between intermediate mass fragments and energetic light particles exhibit characteristic "V" shapes (Fields and co-workers, 1986a) which are qualitatively similar to those measured for two coincident light particles, see Fig. 18a. Since light particles are preferentially emitted in the entrance channel scattering plane (Tsang and coworkers, 1984b; see also Fig. 13), one may conclude that intermediate mass fragments, too, are preferentially emitted in thls plane and that both mean field dynamics as well as statistical aspects are relevant for the interpretation of complex fragment emission In intermediate energy nucleus-nucleus collisions. Figure 28 shows the dependence of the ratio of in-plane to out-ofplane coincidence cross sections, a (n~-180o)/o ~(n¢=90o), on the element number of the coincident intermediate mass fragments, where n~ ¢ -¢. (Fields and co-workers, 1986a). Thls ratio increases with increasing mass of the coincident Y pa~tlcles and wlth increasing particle energy. For correlations between high energy nitrogen ions and alpha particles, the in-plane to out-of-plane ratio is nearly 10.

Nucleus

- Nucleus

Collisions

69

From a d e t a i l e d analysis of the magnitudes of the singles and coincidence cross sections, the following qualitative conclusions were drawn for the '~S+Ag reaction at E/A=22.5 MeV (Fields and co-workers, 1986a): I n t e r m e d i a t e mass fragments are produced in reactions in which more than 20% of the incident linear momentum is carried away by non-equilibrlum particle emission. Particles are preferentially emitted in the entrance channel scattering plane. Non-equilibrlum light particles are emitted with a mean velocity which is directed close to the beam axis and w h i c h is somewhat less than half of the beam velocity. About 100 MeV of energy is dissipated into the random velocity components of nonequilibrlum light particles. The reactions are highly inelastic: a total amount of energy between 200 and 400 MeV is dissipated into internal degrees of freedom of the emitted nuclei. The emission of intermediate mass fragments already sets in during the early, non-equillbrated stages of the reaction. The fragment multiplicities are low, of the order of I, i n d i c a t i n g that i n t e r m e d i a t e mass fragments are emitted with m o d e s t p r o b a b i l i t i e s from a class of reactions r e p r e s e n t i n g about 2/3 of the total reaction cross section. There is a non-negligible probability for the emission of a second fragment.

I

I

I

I

I

Ag(mS, XY) o E / A = 2 2 . 5 M e V o @,-gO* . Ov-40" ¢ +~=~" . 0~-~* • @¥=180". |t.-40" * @r-160". It--?o"

108

107 ="

=

,Y P

10 6

105 I

I

I

10 7

I06

o o

~ -

o

•

!i

%0

105 o

.

.

.

.

I

50

.

.

.

.

I

*

.

.

.

100 0 Energy

,

I

50

.

.

.

.

I

.

100

(MeV)

Fig. 27. Energy spectra of protons (upper part) and alpha particles (lower part) measured in c o i n c i d e n c e with l i t h i u m (left hand side) and carbon fragments (right hand side) emitted in ~2S induced reactions at E / A = 2 2 . 5 M e V . T h e l i g h t particles were detected at the polar angles of % =40 ° (circles) and 8.=70 ° (diamonds) and the azimuthal a~gles of @ -90 ° (ope~n points) and $ =180 ° (solid points); the lithium and carbon fragments wer~ detected at 8 - 27.5 °, # -0 o . The solid lines show the shapes of theAinclusive l~ght particle spectra. (Fields and co-workers, 1986a)

6.3. Liquid-gas phase transition The n u c l e o n - n u c l e o n i n t e r a c t i o n is r e p u l s i v e at short distances and a t t r a c t i v e at large distances. This behaviour is qualitatively similar to the molecular interaction of substances which exist in liquid and gaseous phases, a simple example being the van der Waals gas. Based on such c o n s i d e r a t i o n s , nuclear matter, too, is expected to exist in the liquid and gaseous phases. Early investigation of the nuclear matter e q u a t i o n of state (Sauer and co-workers, 1976; Lamb and co-workers, 1978; Danielewicz, 1979; Schulz and co-workers, 1982; Curtln and coworkers, 1983; Bertsch and Siemens, 1963; Jaqaman and co-workers, 1983, 1984) were motivated, in part, by its astrophysical implications. The search for possible experimental signatures of

70

Claus-Konrad Gelbke and D a ~ d H. Boal

the liquld-gas phase transition research - and controversy.

in nuclear

collision

experiments

is a subject

of o n g o i n g

The essential features of the llquld-gas phase diagram of nuclear matter can be calculated by approximating the nucleon-nucleon force by a zero-range Skyrme-type interaction (Lopez and Siemens, 1984; Boal and Goodman, 1986):

V12= ( - t o + ~

p([~1+r2]/2) ) 6(r I- ~2) ,

t3

(32)

where p denotes the Oenslty and the parameters to and t 3 are chosen to fit the properties cold nuclear matter.

I

I

I

I

I

10 - e:~.x/A>5MeV

--

Xv>Et

I

I

8x = 27.5 °

Y=d

Y=p 0o oo ..°i .-

I

I

A~(S'S,XV) E / A = 2 2 . 5 MeV

o:gx/A>iO UeV Ey>Et+20MeW

I

of

I

00o

Oio lOl Q

II

I

I

I

I

¢ Y=a

Y=t

O

0 o • o @e o • @I

dO 00 OOOOOo 1

I

I

I

I

024-6

-

"

I

I

I

1

8 0 2 4 6 8

Zx Fig. 28. Ratio of in-plane to out-of-plane

c o i n c i d e n c e cross sections between light particles (y = p, d, t, ~) and intermediate mass fragments of element number Z for S2S induced reaction on Ag at E/A=22.5 MeV. LightXparticles were detected at 8.-40 °, intermediate mass fragments were detected at 8.-2~.5 °. Energy thresholds are given in the figure (E t. Ig, 20, 20, 40 MeV for p, d, t, s). Open points represent higher energy thresholds. (Fields and coworkers, 1986a)

The p r e s s u r e P can be expressed interaction term, Pint: Pkin " - V

as the sum of two terms,

dE D(E) £n[ I- f(E))

a kinetic

term, Pkln'

and an

(33)

Q

and

Pint" - ~ to 02 + ~ t3 p 3

(34)

where f(E) denotes the phase space occupancy and

D(E) - g

2~V(2m) 3/2 ( E - Eo ) 1 / 2

h3

(35)

The interaction energy Eo is defined by E0 " - ~ to P + I~ t3 p3 .

(36)

-

Nucleus

Collisions

Nucleus

71

Choosing v a l u e s of t ^ - 782 MeV f ~ 3 and t 3- 10666 MeV fm 6, one o b t a i n s

the pressure

versus

density diagram shown in Fig. ,. (With this choice of parameters, the compressibility of the nuclear fluid is K=229 MeV.) Maxwell's construction can then be used to define the llquld-gas coexistence curve (LGC); the isothermal (ITS) and isentropic (IF.S) splnodal curves are defined by the conditions (BP/~p)T-O and (BP/~p)~=O, respectively. The resulting phase diagram Is shown in Fig. 29. Also Included in the figur~ are curves of constant entropy. The boundaries of the phase diagram are not particularly sensitive to the specific choice of parametrlzatlon for the nucleon-nucleon interaction; rather similar results were obtained by Lopez and Siemens (1984) and Boal and Goodman (1986). However, the inclusion of the Coulomb interaction for finite nuclear systems lowers the predicted value of the critical temperature significantly (Jaqaman and co-workers, 1984; Bonche and co-workers, 1985; Levlt and Bonche, 1985; Reinhardt and Schultz, 1985). For a more detailed discussion of the phase diagram of nuclear matter, the reader is referred to the paper of Goodman and co-workers (1984).

I

;

I'

4.0 7 i

12

i/

,

I ,

i

t

/3.5

,

'

i

,'3.0

,

•.

/~2.5

,

s

,

/

,

•'2.0

..

,

•/

I

, -

~:/~,,li i'-

.~'

\.__

1.5

\-

_---

_.

4

S/A O 5

O

0.2

0.4

0.6

0.8

1.0

1.2

P/Po Flg. 29. Phase diagram of neutral nuclear matter. Isentropes of given values of S/A are shown by the dashed curves. The l i q u i d - g a s coexistence curve is labelled as LGC; the isothermal and Isentroplc splnodals are labelled as ITS and IES, respectively. (Boal and Goodman, 1986) For isothermally expanding systems, the region of mechanical instability Is bounded by the isothermal spinodal curve. If the expansion is isentropic, the region of mechanical instability is bounded by the Isentropic spinodal. The region of mechanical instability is smaller for isentroplc expansion than for isothermal expansion. Intra-nuclear cascade calculations indicate that the expansion of highly excited nuclear systems formed in nucleus-nucleus collisions is nearly Isentropic, see also the discussion of Flg. 7. The comparison of the entropy values of S/A-2-3 given in Fig. 25 with the phase diagram shown in Fig. 29 indicates that nuclear systems f o r m e d In intermediate energy nucleus-nucleus collisions may expand into the region of mechanical instabilities or even pass near the critical point. Indeed, the characteristic power law dependence of the mass distribution shown in Fig. 22 was interpreted as due to statistical cluster formation (Fisher, 1967) near the critical point of the liquld-gas phase diagram of nuclear matter (Finn and co-workers, 1982; Minich and co-workers, 1982; Hirseh and co-workers, 1984; Panagiotou and co-workers, 1984, 1985; Siemens, 1983). H o w e v e r , s e v e r a l i n v e s t i g a t i o n s have shown that the power law dependence of the mass distributions does not provide a unique signature for a liquld-gas phase transition near the critical point. Fragment distributions with power-law dependences can arise from entirely different reaction mechanisms. The gross features of the experimental m ~ s distributions have been interpreted in terms of the random disintegration Of nuclei (Alchelin and HUfner, 1984; Aichelin and co-workers, 1984; Bohrmann and co-workers, 1983) which might be connected with the development of mechanical Instabilities (Bertsch and Mundlnger, 1978; Bertsch and Siemens, 1983), the time-dependent accretion of nucleons from a hot gas of nucleons (MekJian, 1977; Boal, 1983), simple Coulomb barrier penetration effects (Boal, 1984a), or the sequential fragmentation of highly excited expanding nuclear systems (Biro and Knoll, 1985). B o t h

72

Claus-Konrad Gelbke and David H. Boal

equilibrium thermodynamical models (Randrup and Koonin, 1981; F~i and Randrup, 1982; Bondorf and co-workers, 1985a, 1985b; Gross and co-workers, 1982, 1985; Ban-Hao and Gross, 1985) and statistical percolation models (Bauer and co-workers, 1985, 19860) can reproduce the mass distributions from nuclear fragmentation reactions; the connection between these latter two models was investigated by Barz and co-workers (1986) who related the percolation parameter to the excitation energy per nucleon of the fragmenting system. Alternatively, the statistical m o d e l of c o m p o u n d n u c l e a r decay has been generalized to account for the emission of intermediate mass fragments (Moretto, 1975; Friedman and Lynch, 1983a, 1983b). Campi and cow o r k e r s (1984) s u g g e s t e d that the emission of intermediate mass fragments is due to multlfragmentation processes at high energies and due to evaporative processes at lower energies. From the experimental point of view, it has become clear that the shape of the inclusive mass distributions does not contain sufficient information to discriminate between these different theoretical approaches. In fact, there appears to be a qualitative similarity between the gross features of the mass distributions resulting from physically very different processes ranging from the fragmentation of rocks on the macroscopic scale to the fragmentation of nuclei on the microscopic scale (HOfner and Mukhopadhyay, 1986). Biro and co-workers (1985) c o n c l u d e d that the u n i v e r s a l b e h a v l o u r w h i c h seems to characterize mass spectra of fragmentation phenomena was almost independent of the interaction among the constituents and could be qualitatively attributed to the geometrical constraints resulting from the finite available space. Significant progress can only be expected from more detailed experimental and theoretical investigations which go beyond the desoriptlon of inclusive mass distributions. The existence of the phase transition might ultimately be detected only from more subtle effects such as the generation of additional entropy (Lopez and Siemens, 1984; Boal and Goodman, 1986; Csernal, 1985; Csernai and Kapusta, 1986; Kapusta, 1984).

6.4. Towards microscopic descriptions of fragmentation Microscopic descriptions of nuclear fragmentation reactions require the calculation of A-body distributions and fluctuations. On the quantum mechanical level, this task is exceedingly difficult. Progress has been made by introducing fluctuations into the TDHF approach (Knoll and Strack, 1984; Straek and Knoll, 1984). Within this approach, multiplicity distributions and mass spectra have been calculated for isolated nuclear systems evolving from given initial temperatures and densities. The calculations confirm the qualitative expectation of increasing particle multiplicities and the disappearance of large compound nuclear residues at higher temperatures. At present, these calculations have been limited to two spatial dimensions. The development of full three dimensional calculations was initiated at the classical level. As an extension of the intranuclear cascade approach, the nucleon-nucleon interaction can be included via classical two-body potentials. Early calculations of this type were focussed on the prediction of proton energy spectra (Bodmer and co-workers, 1980). The extension of such calculations to fragment formation is hampered by the neglect of the Pauli principle. Wilets and co-workers (1977) suggested that one approximate the effects of the Pauli exclusion principle in terms of momentum dependent potentials which were strongly repulsive in regions of high phase space density. This model has not yet been applied to the fragmentation problem. The qualitative features of the breakup of highly excited finite systems near the critloal region were investigated by Vlncentini and co-workers (1985) by means of computer simulations for a hot classical gas consisting of argon atoms interacting via a Lennard-Jones type of potential. These calculations showed that the reaction trajectories were influenced by the presence of the splnodal curves. Alternative approaches to incorporate the effects of nuclear binding were based on density dependent potentials. These calculations correspond to extensions of the methods which were developed for the numerical solution of the Boltzmann-Uehling-Uhlenbeck equation. The first such calculations for nucleus-nucleus collisions were based on a hybrid approach: The approach and interpenetration phase of the reaction was treated in terms of the intranuclear cascade model in which binding energy effects are neglected. At the end of this phase, the collision term was turned off. The phase space fluctuations generated during the first stage of the reaction were then propagated on an event-by-event basis by replacing each nucleon with a n u m b e r of test p a r t i c l e s w h o s e p o s i t i o n s and m o m e n t a were spread out with Gaussian distributions centered at the classical position and momentum vectors of the corresponding nucleon. The resulting distribution of test particles was then propagated by solving the Vlasov equation (Eq. 19). A density dependent Skyrme-type interaction was used: U(p) = (-124(p/pc) + 70.5(p/pc) 2) SeV .

(37)

Nucleus

-

Nucleus Collisions

73

o)

(b)

Fig. 30. Contour plots of coordinate space density the reaction plane for a single collision nuclei at E / A = 4 0 0 MeV. The m e a n f i e l d i n c l u d e d in part a and o m i t t e d in part Gupta, 1985)

p r o j e c t e d onto between two A=20 i n t e r a c t i o n is b. (Gale and Das

The effect of i n t r o d u c i n g the m e a n field is dramatic. Figure 30 shows the projection of the coordinate space density onto the reaction plane for a single collision of two A-20 nuclei at a b o m b a r d i n g e n e r g y of E / A = 4 0 0 MeV. The simulation was stopped after the main collision epoch while the system was still expanding. Part a of the figure shows the result of a c a l c u l a t i o n for w h i c h the m e a n f i e l d was included; part b shows the result of a calculation for which it was omitted. Clearly, the mean field is essential to preserve density fluctuations and p r o d u c e bound clusters. The approach was recently expanded by several groups who included collisions and the effects of the Pauli e x c l u s i o n p r i n c i p l e (Beauvais and co-workers, 1986; Gr~goire and co-workers, 1986; Aichelin and StOcker, 1986; Bauer and co-workers, 1986d). With these generalizations, the first p h a s e of the c o l l i s i o n can be t r e a t e d as well, eliminating the need for the hybrid approach discussed above. Gr~golre and co-workers (1986) have investigated reactions at r e l a t i v e l y low e x c i t a t i o n e n e r g i e s for w h i c h the f l u c t u a t i o n s must be known with rather good numerical accuracy. To e n s u r e this n u m e r i c a l accuracy, these a u t h o r s a d o p t e d the p r e s c r i p t i o n of r e p r e s e n t i n g each n u c l e o n by a number of test particles whose phase space distribution was spread out by means of Gausslan distributions. Collisions and the Pauli e x c l u s i o n p r i n c i p l e were incorporated as in Eq. 25. This approach requires large amounts of computing time for the simulation of a s i n g l e event. At the time of writing, i n s u f f i c i e n t s t a t i s t i c s have been c o m p i l e d to warrant detailed comparisons with data. The calculations have, however, reproduced several qualitative features of nucleus-nucleus collisions at incident energies of a few tens of MeV per nucleon, such as the onset of incomplete fusion and the energy dependence of linear momentum transfer in fuslon-like reactions. B e c a u s e of the increasing importance of nucleon-nucleon collisions at higher energies, larger volumes of phase space are populated. At these energies, the n u m e r i c a l a c c u r a c y r e q u i r e m e n t s for the c a l c u l a t i o n of f l u c t u a t i o n s are reduced. By using only a single test particle per nucleon (spread out in phase space with a Gaussian distribution), the computation times can be r e d u c e d c o n s i d e r a b l y (Beauvals and c o - w o r k e r s , 1986) and m o r e d e t a i l e d c o m p a r i s o n s with experimental data become feasible.

74

Claus-Konrad Gelbke and David H. Boal

I-

A=40

t~i~mKmON + A=40

E = 100 A ' k~V A=I 0.5-

•

.

.', "+i":,'~"'. .... ,~.-~'~,~..~

. . " .,~.;11"~: ~ - ~- , , ' . ~ ; ~ , '

~.. ;z ~ "

A=4

U

-4

•

0.$-

>

":: :..

A--~ 0.5.

J~ o

-0.5

'-0

0~5 Y

Flg. 31. S c a t t e r plot of the p e r p e n d i c u l a r velocity vs. rapidity for mass A=I, 4, and 35 clusters produced in a s i m u l a t i o n of C a + C a c o l l i s i o n s at E / A = I O 0 MeV. (Beauvals and coworkers, 1986)

(A'40)÷(A-40) 1O0 A'MeV

I

10

20 MASS

30

Fig. 32. A v e r a g e impact parameter associated with a given range of f r a g m e n t mass as c a l c u l a t e d for C a + C a c o l l i s i o n s at E / A - t 0 0 MeV. The bin size c o r r e s p o n d s to 5 mass units. (Beauvais and co-workers, 1986)

Nucleus - Nucleus Collisions

75

As an example, we discuss results obtained for Ca+Ca collisions at E/A-100 MeV. The collisions were f o l l o w e d for a d u r a t i o n of 150 fm/c, by w h i c h time most f r a g m e n t s had s e p a r a t e d in c o o r d i n a t e space. B o u n d c l u s t e r s were d e f i n e d by l i n k i n g groups of n u c l e o n s w h i c h were separated by less than 3 fm from their nearest neighbors. Figure 31 shows a scatter plot of the d i s t r i b u t i o n of the perpendicular velocity vs. rapidity for three fragment masses, A=I, 4, 35. The distributions were averaged over impact parameter. The distribution of nucleons, s h o w n in the upper part of the figure, a p p e a r s r e a s o n a b l y thermal; a large number of nucleons is distributed around the mld-rapidlty region. Fragments of mass A=4 tend to be m o r e s e g r e g a t e d a r o u n d the target and projectile rapldlties; heavier fragments are clearly concentrated around these two rapidity values. This behaviour Is reminiscent of the p a r t l c l p a n t - s p e c t a t o r p i c t u r e in w h i c h l a r g e fragments emerge mainly from peripheral collisions; central collisions, on the other hand, tend to destroy the nuclei. This i n t e r p r e t a t i o n of the c o m p u t e r s i m u l a t i o n is c o r r o b o r a t e d by the r e s u l t s shown in Fig. 32. This figure shows the average impact parameter associated with a given range of fragment masses; heavier fragments are clearly associated with larger impact parameters. In the simulations shown in Flg. 32, the Coulomb interaction was included; a p h e n o m e n o l o g i c a l isospin dependent mean field was used: U(p) = -124 (P/Po) + 70.5 (p/po) 2 + 25 e (pp- pn)/Po MeV

,

(38)

where E = +I ( - I ) f o r p r o t o n s ( n e u t r o n s ) . This ch'olce o f mean f i e l d makes i t possible t o calculate isotopic distributions, see Fig. 33. The calculations predict isotopic d i s t r i b u t i o n s w h i c h have a maximum for N=Z and which extend more to the neutron rlch side than to the proton rlch side. Thls qualitative b e h a v i o u r is o b s e r v e d e x p e r i m e n t a l l y . M o r e i n t e r e s t i n g l y , the calculations p r e d i c t that the " c h e m i c a l " t e m p e r a t u r e s w h i c h c h a r a c t e r i z e the i s o t o p i c distributions should be smaller than the "kinetic" temperatures which c h a r a c t e r i z e the e n e r g y s p e c t r a of the e m i t t e d p a r t i c l e s , a g a i n In q u a l i t a t i v e a g r e e m e n t w l t h w i t h experimental observations (see, e.g. Hirsch and co-workers, 1984).

N 0 0

2

4

6

8

10 NUMBER PER EVENT [ ] >1 [] 0.1-1.0 [] 0.01-0.1 [] 0.001-0.01

N 6 8 10 Flg. 33. I s o t o p i c yields of light fragments predicted by numerical s i m u l a t i o n s for C a + C a c o l l i s i o n s at E / A - t O 0 M e V . T h e calculations were averaged over impact parameter. (Beauvais and co-workers, 1986) N u m e r i c a l simulations may be particularly useful in elucidating the relation between the final reaotlon p r o d u c t s (which are d e t e c t e d e x p e r i m e n t a l l y ) and the t e m p o r a l e v o l u t i o n of the r e a c t i o n (which Is not d i r e c t l y a c c e s s i b l e to e x p e r i m e n t a l investigations). Of particular interest is the q u e s t i o n of w h e t h e r and how the r e a c t i o n is i n f l u e n c e d by the l i q u l d - g a s c o e x i s t e n c e region. For example, it is not clear whether the system vaporizes during the early expansion stage of the reaction and whether fragments condense from the vapor at the late stage of the reaction or, alternatively, whether the system reacts more like a superheated fluid from

PPNP-F

76

Claus-Konrad Gclbkc and David H. Boal

(A=40)+(A=40) 100 A.MeV CENTRAL COLLISION

10~ A=I ~

E

~ 10~

10~

I

I

50

100 ~ME(fm/c)

Flg. 34. Temporal evolution of the average local coordinate space density in the vicinity of nucleons finally emitted as free nucleons (A=I) or in bound clusters (A=4, 20). The simulation corresponds to central Ca+Ca collisions at E/A=IO0 MeV. (Beauvals and co-workers, 1986)

(A-40)÷(A-40)

1.0

1 O0 A.MeV CENTRALCOLUglON

A-20

0.5

E

~ME (fro/el

Flg. 35. Temporal evolutlon of the average p~ase space density in the vicinity of nucleons finally emitted as free nucleons (A=I) or in bound clusters (A=4, 20). The simulation corresponds to central Ca+Ca collisions at E/A-100 MeV. (Beauvals and co-workers, 1986) whloh fast nucleons emerge. Thls latter picture would be more in analogy with bubble formation and the growth of mechanical instabilities. Wlth the help of computer simulations one may address such questions by investigating the time dependence of the local coordinate and phase

Nucleus

- Nucleus

Collisions

77

s p a c e densities in the vicinity of individual nucleons. At the end of the collision, it can be determined whether an individual nucleon has emerged as a free particle or as part of a bound cluster. By this m e t h o d one may d e t e r m i n e w h e t h e r clusters have condensed from low density (gaseous) regions or whether nucleons in clusters have always remained in high density (liquid) r e g i o n s . The results of such an analysis (Beauvais and co-workers, 1986) are given in Fig. 34. The figure shows the temporal evolution of the average local coordinate space density for free nucleons, and nucleons finally emerging in clusters of mass A-4 and 20. After the initial phase of the reaction, t~30 fm/c, the density begins to decrease as the s y s t e m expands. The s y s t e m e x p a n d s to about half normal nuclear matter density and then breaks up into bound clusters and nucleons. N u c l e o n s b o u n d in c l u s t e r s do not o r i g i n a t e from r e g i o n s of low d e n s i t y w h i c h c o n t r a c t into r e g i o n s of higher d e n s i t y as c l u s t e r s are formed. This numerical simulation suggests that fragment formation is not similar to droplet condensation f r o m a dilute g a s e o u s p h a s e but r a t h e r to the b r e a k u p of n u c l e a r m a t t e r due to the d e v e l o p m e n t of m e c h a n i c a l instabilities. T h i s i n t e r p r e t a t i o n is s u b s t a n t i a t e d by the t e m p o r a l e v o l u t i o n of the average phase space densities shown in Fig. 35. While the coordinate space densities are very s i m i l a r during the e a r l y v i o l e n t s t a g e s of the reaction, the phase space densities are not. Particles which will ultimately emerge in the form of free nucleons scatter into regions of low phase space d e n s i t y at a very early stage of the reaction. Particles which will ultimately emerge in the form of clusters remain in regions of relatively high phase space density throughout the reaction. This behaviour bears qualitative resemblance to a percolation mechanism. T o s u m m a r i z e , m i c r o s c o p i c f o r m u l a t i o n s of n u c l e a r f r a g m e n t a t i o n have been developed s u f f i c i e n t l y far at the s e m i c l a s s i c a l level that c o m p a r i s o n s w i t h e x p e r i m e n t a l data are becoming feasible. The calculations already reproduce many of the qualitative features observed experimentally; more quantitative tests are expected to be available soon. It appears that the evolution of the reactions is influenced by the presence of mechanical instabilities; however, c l e a r s i g n a t u r e s of the liquid-gas phase transition which could be tested experimentally have not yet emerged.

7. POPULATION OF EXCITED STATES For intermediate energy nucleus-nucleus collisions, particle e m i s s i o n a l r e a d y sets in d u r i n g t h e e a r l y n o n e q u i l i b r a t e d stages of the reaction. In the a b s e n c e of d e t a i l e d d y n a m i c a l treatments of the spatial and temporal evolution of the collision process, r e c o u r s e is o f t e n t a k e n to m o d e l s b a s e d on the assumption of statistical particle emission from nuclear subsets c h a r a c t e r i s e d by their a v e r a g e velocity, s p a c e - t i m e e x t e n t , a n d e x c i t a t i o n e n e r g y or " t e m p e r a t u r e " (Aiehelin, 1984; Fields and co-workers, 1984; Gosset and co-workers, 1978; Jacak and co-workers, 1983; Knoll, 1979; S h y a m and Knoll, 1984a, 1984b, 1986; W e s t f a l l and cow o r k e r s , 1976). It is c l e a r l y i m p o r t a n t to p e r f o r m e x p e r i m e n t s which test the validity of statistical concepts and which are sensitive to the space-time e v o l u t i o n of the reaction. Of p a r t i c u l a r i m p o r t a n c e are m e a s u r e m e n t s of the a v e r a g e e x c i t a t i o n energy per n u c l e o n or "temperature". It was already discussed in previous sections, that the energy spectra of particles emitted at mid-rapidities can be rather well described in terms of simple M a x w e l l i a n d i s t r i b u t i o n s , the t e m p e r a t u r e p a r a m e t e r s of which are nearly independent of the emitted fragments (Westfall and c o - w o r k e r s , 1982; J a c a k and c o - w o r k e r s , 1983). This o b s e r v a t i o n is c o n s i s t e n t w i t h t h e assumption t h a t t h e v e l o c i t y d i s t r i b u t i o n s of the e m i t t e d p a r t i c l e s are c l o s e to the equilibrium limit. Most attempts to obtain experimental information about the t e m p e r a t u r e of highly excited nuclear systems were, therefore, based on analyses of the kinetic energy spectra of the emitted particles, see also Fig. 11. H o w e v e r , the i n t e r p r e t a t i o n of s i n g l e - p a r t i c l e i n c l u s i v e k i n e t i c energy spectra can be complicated by their sensitivity to collective motion (Siemens and Rasmussen, 1979), the temporal e v o l u t i o n of the e m i t t i n g s y s t e m ( F r i e d m a n and Lynch, 1983a, 1983b; Fields and co-workers, 1984; StOcker and co-workers, 1981), the sequential decay of highly excited primary fragments (Friedman and Lynch, 1983a, 1983b), and f l u c t u a t i o n s of the Coulomb barrier (Ban-Hao and Gross, 1985). Information about the excitation energy per nucleon or "temperature" at the point at w h i c h the p a r t i c l e s leave the equilibrated (sub)system can be obtained from the relative populations of nuclear states (Morrlssey and c o - w o r k e r s , 1984, 1985; P o c h o d z a l l a and c o - w o r k e r s , 1985a, 1 9 8 5 b ) . In m o s t s t a t i s t i c a l c a l c u l a t i o n s , i n - m e d i u m c o r r e c t i o n s are n e g l e c t e d and the asymptotic nuclear states (bound and unbound) are used for the specification of the a v a i l a b l e decay configurations. S u c h c a l c u l a t i o n s m a k e s p e c i f i c p r e d i c t i o n s a b o u t the r e l a t i v e

78

Claus-Konrad Gelbke and David H. Boal

populations of ground and excited states of the u n i q u e r e l a t i o n exists between the relative Determinations of emission temperatures from the if the emitting subsystem is not only close to equilibrium.

emitted fragments. Indeed, in each model a population of states and the temperature. relative populations of states are meaningful kinetic equilibrium but also close to chemical

7.1. Population of particle stable states Relative populations of ground and low lying (particle stable) excited states of light nuclei were investigated by via particle-X coincidence techniques (Morrissey and co-workers, 1984, 1985, 1986; Bloch and co-workers, 1986; Xu and co-workers, 1986). For ~ N induced reactions, surprizingly few 7Li, eLi, and 'Be nuclei were observed to be emitted in particle excited states relative to the respective ground states (Morrissey and co-workers, 1984, 1985). This observation was interpreted to imply emission temperatures T$I MeV, significantly smaller than either the temperature of the compound nucleus or the temperature parameters which charactsrlze the kinetic energy spectra of these light nuclei (Morrissey and co-workers, 1984, 1985). For an expanding system, the "kinetic" temperature, measured by the slope of the particle energy spectra, can be different from the "chemical" temperature, measured by the relative population of states (Boal, 1984b). The kinetic temperatures are expected to remain nearly constant for adiabatic expansions, since they are closely related to the energy of the system. The chemical temperatures, on the other hand, are defined in a frame comoving with the expanding gas or liquid; as a consequence, they are expected to decrease during an adiabatic expansion until the system goes out of equilibrium. Based on such general considerations, one may expect that chemical temperatures should be smaller than kinetic temperatures. The exact magnitude of this difference should depend on the densities at which the various degrees of freedom go out of (local) equilibrium; at present these densities are only poorly known. Unfortunately, temperatures deduced from measurements of relative populations of states can have large errors whenever the primary population ratios are altered by secondary processes (Pochodzalla and co-workers, 1985a). For the case of chemical equilibrium, the p r i m a r y population ratio of two states of spins S~ and S 2 and level separation AE-E2-E , is given by: Rp- (2S2+I)(2S,+I)exp(-AE/T) - R®exp(-~E/T)

.

(39)

If this ratio is altered by an unknown factor, ~, and if the resulting ratio is used to determine the temperature T' from the equilibrium relation, sR =R exp(-aE/T'), one obtains T'-T/(I-£n(s)T/aE). Only for the limit T{£n(a){<>AE, one obtains T'-{~m/£n(a)[, independent of the temperature T. If one cannot ensure that s is close to unity (or else that the value of e is preolsely known), the population ratios can only be expected to be useful for the e x t r a c t i o n of temperatures for T
79

Nucleus - Nucleus Collisions

(The upper and lower solid curves show calculations for densities of 0.3p and 0.7p , where p o denotes the density of normal nuclear matter.) The calculations predict considerable reductzo~ of the final fractions, F ; for TZ4 MeV the predicted fractions are close to the experimental values. Xu and co-workers e~timate an uncertainty of at least 25~ in the theoretical values of ~ due to the present incomplete understanding of fragmentation processes and due to the lack spectroscopic information concerning the spins and branching ratios for many of the tabulated particle unstable states. Theoretical uncertainties and the insensitivity of the theoretical values of F to temperature prevent the extraction of m e a n i n g f u l e m i s s i o n temperatures, for TZ2 MeV~ from these low energy Y-ray transitions (Xu and co-workers, 1986). 0.50

100

(0.051 -. g.,.) 0.25

50

400

8oo J,~ A LaJa~

0.00

zooo 'Be

7Be . (.0...4:.~.9. . . . -~ g.s.) _ _

0.25

~-00 0.00

300

o {3~-00 . . . .

400

500

i

i

50

.

-

. . . .

~

i

"''1

. . . .

i . . . .

i ' ' "

•

*'s (0.953 --, g.,.)

~ I . . : . i . . .

I . . i - : -

. i . i ;

i : : : i

~sc (s.854 -, s.e84)

0.0

400 0.1 :

~

/

, o o , , o

~00

lo0

~oo E (keV)

2

4

6

8

0.0

10

T (MeV)

Fig. 36. Left hand side: Doppler shift corrected X-ray spectra measured in coincidence with SLl, ~Be, *°B, *2B, and *3C nuclei produced in '2S induced reactions on Ag at E/A=22.3 MeV. The dotted lines show background spectra. Right hand side: Fractional probabilities, Fy, that an o b s e r v e d f r a g m e n t is accompanied by the designated X-ray. The dotted lines indicate the range of values consistent with the coincidence measurements. Solid (dashed) curves are results of s t a t i s t i c a l c a l c u l a t i o n s for w h i c h the p o p u l a t i o n s of p a r t l c i e unbound states are included (neglected). (Xu and co-workers, 1986)

7.2. Population of particle unbound states The effects of sequential feeding from higher lying particle unstable states may be less important for the relative populations of short-lived particle unstable states separated by large energy intervals dE (Pochodzalla and co-workers, 1985a; 1985c). Populations of particle unstable states in light nuclei were measured for *~N induced reactions on *'TAu at E/A=35 MeV (Chltwood and co-workers, 1986b) and *°At induced reactions on l''Au at E/A=60 MeV (Pochodzalla and co-workers, 1985a, 1985c, 1986c) using a close-packed array of 13 AE-E telescopes. In order to suppress structures caused by the granularity or finite phase space acceptance of the experimental apparatus, it is convenient to construct the correlation function, R(q), which is defined in terms of the coincidence yield, Yi2(Pl,P2 ),_ and the single particle yields, YI(Pl )_ and Y 2 ( ~ 2 ) :

Claus-Konrad Gelbke and David H. Boal

80

Y12(~I,;2) " C12"(I+R(~))'~ YI(~I)'Y2(;2)

(~0)

•

Here, ~I and ~_ are the laboratory ~ o m e n t a of p a r t i c l e s I and 2, and q is the momentum Of relative motlon~non-relativlstlcally q - g(~2/mg-p~I/ml)). The normalization constant, C.^, is determined by the requirement that R(q)-O fo~ l&rge relative momenta. For each experilm~ental gating condition, the sums on both sides of Eq. 40 are extended over all energy and detector comblnations corresponding to t~e÷ given bins of q; in most cases the correlation functions are determined as a function of q = lql only. The coincidence yield, Y , resulting from the decay of excited nuclei can be extracted by assuming that the total ~olncidence yield is given by Y.^- Y + C12.Y.Y211+RB(q)], where Rb(q) denotes the "background" c o r r e l a t i o n f u n c t i o n (~cho~zalla an~ co-worKers 1985a).

F'6Li (MeY) 2

4 1

I0 ÷

t

8 I

'

197Au( 4OAr, -d)X, E/A=6OMeV

%

ec, = 30 °

1.0

i0-~

e•

>. 197Au( 4OAr,tZd ) X

0.6

4e

0"

6 I

'

4(

of

I

0.2

,o4

E/A = 60 oMeV Say = :50

,

L~.=

.

0

50

I00

.... .,.

150 ..... • . . 2 0 , . v

io3

@

\...

,,

@

....

0 ~ 0

2.5 MeV

...... i02

I00

200 q (MeV/c)

300

I

0

",

,u,v I 2

,

Ik

\,

~',

I \ 4

I

\

"K"-.~

/

~L"-~'..J \1

I\~1"." 6

8

Tc.m.(MeV)

Fig. 37. Part a: Correlation function for coincident deuterons and alpha particles measured for ~°Ar+Ig~Au reaction at E/A=60 MeV. The solid curve shows the background correlation function. Part b: Energy spectrum resulting from the decay of partlcle-unstable states in SLi. The curves correspond to thermal distributions with T=I, 2.5, 5, IO, and 20 MeV, taking the efficiency of the experimental apparatus into account. (Pochodzalla and co-workers, 1985a) As an example, Fig. 37a shows the e-d correlation function measured for the ~°Ar+ig~Au reaction at E/A=60 MeV (Pochodzalla and co-workers, 1985a). The correlation function exhibits two maxima which correspond to the T-O state in 'Li at 5.186 MeV (J~- 3 , F = 24 keV, F /F~ L-I.00) a~d the overlapping T=O states at 4.31MeV (JW=2 , F-1.3 MeV, F /F~ --0.97) and a~ 5 . ~ M e V (JW-1 , F-1.9 MeV, F /F t t-O.74). Yields of particle unstable excited ~ nuclei, shown in Fig. 37b, were extracted ~y binning the experimental yield with respect to the kinetic energy in ~he '51 rest frame and subtracting the background yield defined via the background correlation funotlon shown by the solid llne in Fig. 37a. The yield Y^(E ) Is related to the energy spectrum, dn(E)/dE, in the rest frame of the decaying nucleus by ~he equation _. dn(E) .. Yc(E*) - ; ¢ ( E * , g ) . ~ . u ~

(41)

N u c l e u s - N u c l e u s Collisions

81

ff

Here, ¢(E ,E) is the e f f i c i e n c y f u n c t i o n of the experimental apparatus f~r the detection of particle pairs resulting from the decay of particle unstable nuclei; E and E denote the actual and m e a s u r e d excitation energies, respectively. The efficiency function can be determined from Monte Carlo calculations which take the precise geometry and detector r e s p o n s e into account; for more details, see Pochodzalla and co-workers (1985a, 1986c). Information about the emission temperature of the system can be obtained by i n v e s t i g a t i n g the r e l a t i v e populations of states. If the energy dependence of the phase shifts is dominated by a series of resonances of spins, J1' and widths, r i, the thermal p o p u l a t i o n of states decaying into the channel c can be writte~ as dn(E)] . .e-E/T.[ --d'E'-"c N i

(2Ji+1)rl/2~

Zc

(E-Ei)2+F~/4. ri,i

(42) '

where r ./r. denotes the branching ratio for the decay into the channel c. For the case of c I I narrow res6nances, Eq. 42 reduces to the familiar expression r nc(E i) ~ -~.'i.(2Ji+1).exp(-Ei/T) i

,

(43)

from which one can derive the population ratio, Eq. 39. Calculations based on Eqs. 41 and 42 are shown in Figure 37b for a variety of e m i s s i o n t e m p e r a t u r e s . The spectral shapes are sensitive to temperatures smaller than the level separation; higher emission t e m p e r a t u r e s are m o r e difficult to distinguish. The e x p e r i m e n t a l yields are c o n s i s t e n t with an e m i s s i o n temperature of T = 5 MeV. This value is lower than the temperature parameter T = 20 MeV w h i c h c h a r a c t e r i z e s the energy spectra of complex nuclei emitted at intermediate rapidity (Jacak and co-workers, 1983; Pochodzalla and co-workers 1985a, 1986c), see also Fig. 11. More accurate d e t e r m i n a t i o n s of emission temperatures can be made by measuring the relative p o p u l a t i o n s of states s e p a r a t e d by s i g n i f i c a n t l y larger energy i n t e r v a l s , A E > > T . S u c h m e a s u r e m e n t s were p e r f o r m e d for the ~ ° A r + ~ ' ' A u reaction at E/A=60 MeV (Pochodzalla and coworkers, 1985c, 1986c). The solid lines in Fig. 38 show the predicted temperature dependence of the relative yields, NL/NH, from low and high lying states in ~He, SLi, and abe. (The specific states are indicated in-th~ figure; the calculations include the response of the e x p e r i m e n t a l apparatus). The h a t c h e d areas display the measurements of Pochodzalla and co-workers (1985c, 1986c). The experimental yield ratios are consistent with temperatures of T-4-5 MeV. Because of the rapid v a r i a t i o n of the p r e d i c t e d yield ratios, signiflcantly higher temperatures can be excluded - unless secondary processes could lead to a l t e r a t i o n s of the primary p o p u l a t i o n ratios by large factors. Experimental information about sequential decays feeding particle stable light nuclei is a v a i l a b l e for the ~ ° A r + * ' ' A u reaction at E/A-60 MeV (Pochodzalla and co-workers, 1986c). The measured feeding intensities could be rather well described in terms of the quantum statistical m o d e l of H a h n and S t O c k e r (1986). In this model the primary fragment d i s t r i b u t i o n s are approximated in terms of equilibrium distributions for dilute nuclear matter characterized by a t e m p e r a t u r e T and a density p; the calculations incorporate known states of nuclei with AS20. Secondary distributions are obtained by performing statistical model calulatlons for the decay of e x c i t e d fragments. F i g u r e 39 illustrates the effects of feeding from high lying particle unbound states on the relative populations of higher lying states (Pochodzalla and co-workers, 1 9 8 6 0 ) . T h e s o l i d points show the "apparent t e m p e r a t u r e s " e x t r a c t e d from the m e a s u r e d population ratios of s p e c i f i c states by applying Eq. 42, i.e. by n e g l e c t i n g f e e d i n g from secondary decays. Apparent emission temperatures extracted from the relative populations of the ground and low lying (particle unstable) states in 'Li, 7Li and 'Be are inconsistent with those e x t r a c t e d from the r e l a t i v e p o p u l a t i o n s of widely separated states ~He, SLi, 'Be or higher lying particle unbound states in 6Li. These discrepancies can be understood as due to secondary decay processes. In fact, the measured population ratios can be rather well described in terms of the quantum statistical model by assuming a single emission temperature, T=5.5 MeV, and b r e a k u p density p=0.04po , see solid histogram in Fig. 39. These calculations indicate that all measured population ratios may be affected by secondary decays. F i e l d s and c o - w o r k e r s (1986b) investigated the effects of sequential decays by assuming, for simplicity, Dhat each available state, i, is initially populated with the weight, PI' given by Pi ~ (2Si+1)exp( - VI/T - Bi/T ) .

(44)

82

Claus-Konrad Oeibke and David H. Boal

I00 4%r + ~tl,

4Heg.s.

4He~.t

E/A z 60 MeV0 ~ov" 30°

xl(

z~

"- 10 z~

llll,~lil" 0

5

0

I0

5 T (MeV)

5

I0

Fig. 38. Yield ratios, N. /NH, for low and high lying states in ~He, aLl, and aBe. ~he solid curves show the calculated ratios as a function of temperature. The hatched regions show measurements for ~°Ar+*9?Au reaction at E/A=60 MeV. The following decays were investigated: ~He(20.1 MeV) + p+t; 9Li(g.s.) + p+e; SLi(16.66 MeV) ÷ d+SHe; aBe(3.04 MeV) a+a; 'Be(17.64 MeV) ~ p+?Li. (Pochodzalla and co-workers, 1986c)

i

J

4°Ar+igVAu, E/A=60MeV, ear=30 ' ...... QSM: T=5.5

+I

MeV. p/po=O.04 8Be17.e/~3es.o4 ~3e.~.oJ~3el.,. ?Li4.e3/TLi,tabl,

%i4..~+5.,J~Li~.,, -+

, I

eLiz.19/SLi,~ble ~%i,s.ee/SLig.,. ~ io.1/~tl.l.

0

I

i

5

I0

15

APPARENT TEMPERATURE (MeV) Fig. 39. "Apparent" temperatures extracted from the r e l a t i v e populations of specific states via Eqs. 41 and 42. The solid points represent experimental values. The solid histogram s h o w s v a l u e s p r e d i c t e d by the q u a n t u m statistical model once feeding from higher lying states has been taken into account. The distribution of primary fragments was calculated for a temperature of T=5.5 MeV (indicated by the dashed llne) and a density of p=0.O4p 0. (Pochodzalla and co-workers, 1986c) In Eq. 44, S i is the spin of the primary fragment; V. and B i denote the Coulomb barrier and separation en@rgy for emission from a heavy parent n~eleus of temperature T. The binding energies of heavy nuclei were calculated from the Welzs~cker mass formula; for the emitted

83

Nucleus - Nucleus Collisions light fragments (Ai$20), all known states with widths less than 3 MeV were included in calculations. Foff the primary distribution defined by Eq. 44, the population ratio of different states, i and J, of a given isotope is given by Eq. 39, since AE=B.-B. and Vi-V.. decay of particle unstable states to available final states was treated ~n terms ~f statistical model (Hauser and Feshbaoh, 1952).

10 0

: ' ' ' i ' ' ' i "

i

'

'

'

i

'

'

'

!

i

'

i

'

'

'

I

'

'

'

i

the two The the

"

m/A=60 MeV

• 10-1 _ a ( ~ ) x 2

. _6Li(~) ~_" ; %e( a..o4} ..

20.1

•

16.7

17.6

/

•

/

,"

,." ,

,

~

,~

~

J.

:

1 0 - 2 8

~10-3

•

. lJ,'l

"

"

• e r ; (

•

"

I

.

.

.

I .

'

"

"

I

2.1o

"

%

'-'+ %.8'~b!~'

100

.l. "

"

I .

.

.

I .

I

"

"

I

"

' 6 I. i {4.$.6.06% J'~%. 2.19

I .

.

.

I .

I

"

"

I

"'

I

.

'

8Be(S.~ %|.I./

/

y+

Ii

10-1

. :"

:... 10-2

: . . .

, . . I . . . I .

O

4

80

I

4

,

,

.

I

.

8

:,

.

.

I

4

.

,

,

8

T (M v) Fig. 40. Temperature dependence of population ratios, R /R®, for s p e c i f i c states in *He, SLi, 6Li, "Be nuclei. Ratios measured for the ~°Ar+*'?Au reaction at E/A-60 MeV are s h o w n by hatched regions• Dotted curves: temperature dependence of primary population ratios; solid curves: final ratios predicted from Eq. 44; dashed and dasheddotted c u r v e s : final r a t i o s p r e d i c t e d by q u a n t u m statistical calculations for densities of P/Pc= 0.05 and 0.9, respectively. (Fields and co-workers, 1986b) Figure 40 shows the calculated effects of secondary decays on the relative populations of states in several light nuclei. The dotted curves show the temperature dependences of the primary population ratios. The solid curves indicate the final population ratios when one includes the sequential decays of excited primary fragments populated according to Eq. 44. At low emission temperatures, T$I MeV, few unbound states are populated; primary and final population ratios are equivalent, and Eq. 42 (or, for the case of narrow resonances, Eqs. 43 or 39) can be used to determine emission temperatures. At higher temperatures, however, the final populations do not follow the thermal relation of Eq. 43. Instead, they reflect the feeding from higher lying levels. All population ratios are predicted to reach saturation values less than expected from the simple relation of Eq. 39. The calculations corroborate the qualitative arguments given by Pochodzalla and co-workers (1985a) that Eq. 39 can only be used to determine emission temperatures when aE/T<
Claus-Konrad Oclbkc and David H. Boal

84

mass distributions. Mass distributions, calculated from Eq. 44, become flatter at higher temperatures. This increases the modifications of the population ratios. However, differences exist between different statistical distributions of the primary fragments. To illustrate these differences, the dashed and dashed-dotted curves show the results of calculations with the quantum statistical model (Hahn and St~cker, 1986) for densities of plpo= 0.05 and 0.9, respectively. Penetration through the Coulomb barrier is neglected in this model. The predicted mass distributions become flatter with increasing breakup density. As a consequence, the modifications due to sequential decay processes are l a r g e r for higher breakup densities. Further specification of the primary distribution is possible only with additional constraints provided by mass, charge, and isotopic distributions (Fields and co-workers, 1986b).

10 2

l

. . . .

I

. . . .

I

Ap=131,

"W

. . . .

I

. . . .

. . . .

I

Zp---54, T=5 M e V

•

.,i,J

[]

- ~

lOo

10 2

I

. . . .

I

. . . .

I

GeV

• Data, Hirsch et al.

states Bound excited

Stable g.s.

. . . .

Xe(p,X), E p = 8 0 - 3 5 0

• Final _f-- Primary-all

101

I

Y Calculation

(T=5 M e V

"~ 101

,6 10-1

b '~

10 o

10-2

! 10-3

10-1

0

5

10 Ax

15

20

,

0

5

,

,

.

I

. . . .

10

I

15

~

.

.

II

20

Ax

Fig. 41. Part a: Mass distributions calculated from Eq. 44 for emission from a Xe nucleus at T-5 MeV. (Histogram: primary distribution; open points: final distribution; dark and light shaded regions show contributions from bound ground and e x c i t e d s t a t e s , r e s p e c t i v e l y . ) P a r t b: M a s s distribution from proton induced reactions on Xe (solid points) and final distribution predicted from Eq. 44 (histogram). (Fields and co-workers, 1986b) Sequential decays do not only affect the relative populations of states but also the shapes of the mass distributions. Figure 41a shows primary mass distributions calculated from Eq. 44 for emission from a moderately excited Xe nucleus of 5 MeV temperature (Fields and co-workers, 1986b). The histogram represents the total primary mass distribution including unbound states; dark and light shaded regions represent contributions from ground and excited particle stable states; solid points in the figure show the final mass distribution after the decay of particle unstable states. The primary distributions are relatively smooth with little s t r u c t u r e resulting from variations in level density and binding energies; the final mass distributions, on the other hand, exhibit more detailed structure. Figure 41b compares the calculations wlth the fragmentation data of Hirsch and co-workers (1984) for p+Xe collisions at 80-350 GeV energy. The structures in the experimental mass distributions for light nuclei (6&A$14) emitted in fragmentation reactions and solar abundances have been interpreted in terms of a depletion resulting from the in-medium effect of Paull blocking (ROpke, ;983; ROpke and co-workers, 1985). The calculations shown In Fig. 41 indicate that such structures can be largely due to the emission and decay of particle unbound excited nuclei (Fields and co-workers, 1986b). In summary, the emission of unbound nuclei is an essential feature of the disintegration of highly excited nuclear systems. The sequential decay of particle unbound states leads to structures in the mass yields which agree with those observed in high energy fragmentation reactions. At temperatures, T>I MeV, the relative populations of states are strongly affected

Nudeus - Nucleus Collisions

85

by s e c o n d a r y decays. Present experiments begin to shed light on the role of isolated particle u n b o u n d states in nuclear f r a g m e n t a t i o n reactions. However, virtually no e x p e r i m e n t a l information exists on the populations of higher lying, overlapping continuum states.

8. TWO-PARTICLE CORRELATION FUNCTIONS W h e n two particles are emitted in close p r o x i m i t y in space or time, their wave function of relative motion can be modified by final state interactions and quantum statistical symmetries. M e a s u r e m e n t s of t w o - p a r t l c l e c o r r e l a t i o n s at small relative momenta can, therefore, provide information about the space-time characteristics of the emitting system ( H a n b u r y - B r o w n and Twiss, 19561 O o l d h a b e r and co-workers, 19601Shuryak, 19731Kopylov and Podgoretskli, 19741 Kopylov, 19741Cocconi, 19741Koonin, 1 9 7 7 1 Y a n o and Koonin, 1 9 7 8 1 S a t o and Yazaki, 19801 Boal and Shlllcock, 19861Jennlngs and co-workers, 1986). For intermediate and high energy nucleusnucleus collisions, the s p a c e - t i m e e v o l u t i o n of the e m i t t i n g system has been e x p l o r e d by m e a s u r i n g correlations between two pions (Fung and co-workers, 1 9 7 8 1 L u and co-workers, 1981; Beavis and co-workers, 1983a, 1983b; Zajc and co-workers, 1984), two protons (Zarbakhsh and coworkers, 19811 Lynch and co-workers, 1983; Gustafsson and co-workers, 19841Pochodzalla and coworkers, 1986a, 1986c), two deuterons (Chitwood and co-workers, 1 9 8 5 1 P o c h o d z a l l a and coworkers, 1986a, 1986c), two tritons (Chltwood and co-workers, 19851Pochodzalla and co-workers, 1986a, 1986c), protons and alpha particles (Pochodzalla and co-workers, 1986a), and deuterons and alpha particles (Pochodzalla and co-workers, 1985a, 1986a, 1986c; Chitwood and co-workers, 1986b; Chen and co-workers, 1986). The s e n s i t i v i t y of the t w o - p a r t i c l e c o r r e l a t i o n function, R(q), defined by Eq. 40, to the spatial dimensions of the emitting system can be understood most easily in the thermal model (Jennings and co-workers, 1986). For a thermalized system contained in a volume V, the twoparticle density of states can be approximated as: p(P,q)

= po(P).p(q)

,

(45)

where pc(P) = Vp2/2~ 2 denotes the density of states associated with the m o t i o n of the centero f - m a s s of the two particles. The density of states for the relative m o t i o n of the two particles can be approximated as: P(q) = Pc(q) + ~P(q)

(46)

•

Here, the plane wave density of states for particles with spin s i (i=1,2) is given by: Pc(q) =

(2SI+1)(2S2+1).V'q:= . 2~

For non-ldentical particles, ap(q) =

I ~

(47)

the interaction term, Ap, can be written as: a6j

[

'&

(2J+I).

(48)

aq

J,~

Here, 62 o is the scattering phase shift for orbital angular m o m e n t u m £ and total angular momentum'3. In contrast to the plane-wave densities, the interaction term does not depend~nt ~n the size of the system. Introducing the plane wave density of states, p0(p i) = (2si+1).VPi/2. , for particles i = I, 2 one obtains: p(P,q) = po(P)po(q)-[1+R(q))

= po(pl)p0(pm).(1÷R(q)l

,

(49)

where the correlation function is given by: 2~

R(q) =

a6j " X (2J+I). '£ (2s1+1)(2s2+1).Vq 2 J,£ aq

(50)

If one assumes, that the singles and coincidence yields are given by a thermal distribution: Yi(Pi ) = po(pi).e-Ei/T

,

(i - I, 2)

(51)

and: Y12(P1,P2 ) = p(P,q).e

-(E.+E.)/T , ~ ,

(52)

Claus-Konrad Gelbke and David H. Boal

86

one can see immediately that R(q) agrees with the definition of Eq. (40). At this level of approximation for the thermal model, the two-particle correlation function depends only on the volume of the emltting system. It is independent of the temperature of the system. (Information about the temperature of the system Is obtained from the relative populations of states, dn(E)/dE-N.ap(E).exp(-E/T). If the energy dependence of the phase shifts is dominated by a series of resonances, one obtains Eq. 42, see the discussion in the previous section. 1 Historically, the sensitivity of two-particle correlation functions to the space-time extent of the emitting system was derived from the modifications of the plane waves describing the relative motion of the two particles by their respective interactions (Koonin, 1977; Sate and Yazakl, 1980; Jennings and co-workers, 1986) or, for the case of identical particles, by the r e s t r i c t i o n s imposed by quantum statistical symmetries (Hanbury-Brown and Twiss, 1956; Goldhaber and co-workers, 1960; Shuryak, 1973; Kopylov, 1974; Kopylov and Podgoretskli, 1974; Cocconi, 1974; Yano and Koonln, 1978). If the time dependence of the emission process is neglected, one can express + the two-particle correlation function in terms of the single + + particle source function, p(r), and the two-body wave function ¥(r I,r2):

R(q) -

d3rld

r2{l~(r1,~2)I

}.P(rl)P(r2) x { d3r p(~))-2.

This formula was o r l g l n a l l y given by Koonln (1977); Jennlngs and co-workers have recently shown that i t Is equivalent to Eq. 50 derived from the thermal model. Most analyses of t w o - p a r t l c l e correlations performed up to now, were based on the original formulatlo~ o~ Koonln, Rq. 53, and the assumption of a source of Oausslan spatlal density, p(r) ~ e x p [ - r / r o ] , and n e g l l g l b l e lifetime. For fixed spatial source dimensions, the magnitude of two-particle correlations is expected to decrease when the particles are emitted at larger time intervals. The simplifying assumption of negligible lifetime can, therefore, only provide upper limits for the spatial dimensions of the emitting system.

8.1. Two-partlcle correlation functions from inclusive measurements Two-particle correlation functions for intermediate energy nucleus-nucleus collisions were investigated for *60 induced reactions on *'~Au (Lynch and co-workers, 1983; Chltwood and coworkers, 1985) and on '2C, ~ A I (Bernsteln and co-workers, 1985) at E/A-25 MeV, for *~N lnduced reactions on *''Au at E/A-35 MeV (Chltwood and co-workers, 1986b; Pochodzalla and co-workers, 1986a, 1986b; Chen and co-workers, 1986), and for ~°Ar lnduced reactions on ~''Ar at E/A-60 MeV (Pochodzalla and co-workers, 1986c; Kyanowski and co-workers, 1986). Figure 42 shows the two-proton correlation function measured for *~N induced reactions on *'~Au at E/A-35 MeV and an average emission angle of ®a.-35 ° (Pochodzalla and co-workers, 1986a). The attractive slnglet S-wave Interactlon between ~he two colncident protons glves rise to a maximum in the correlation function at a relative momentum of q-20 MeV/c. This maximum becomes larger with Increasing total kinetic energy of the two colncident particles lndlcatlng that more energetic light particles are emitted from sources whlch are more localized In space-time. A more quantitative scale is provided by the solid, dashed and dotted curves which represent theoretical correlation functions predicted by the final-state interaction model of Koonln (1977) for a source of Gaussian spatial density, p-po.exp(-r2/r:), and negligible lifetime, see Eq. 53. As an example of correlations between non-ldentlcal particles with resonant interactlons, Fig. 43 shows e-d correlation functions measured for the *~N+*97Au reaction at E/A-35 MeV (Chitwood and co-workers, 1986b). As was already discussed in connection with Fig. 37a, the s-d correlation functlon exhibits a narrow peak corresponding to the 2.186 MeV state in "Ll and a broader peak correspondlng to the overlapping states at 4.31 and 5.65 MeV. The measured correlatlon functions exhiblt only a slight angular dependence. In qualitative agreement with the trend observed for two-proton correlations, the measured correlations b e c o m e m o r e pronounced with Increaslng kinetic energles of the coincident particles. The solid and dotted lines in the figure show theoretical correlation functions (Boal and Shillcock, 1986; Chltwood and co-workers, 1986b) predicted by the final-state Interaction model. Signlflcant correlations are also observed between two particles for which the Interactlon does not exhlblt resonant features. As an example, Flg. 44 shows two-deuteron correlation functions measured for ~°Ar induced reactions on *''Au at E/A-60 MeV (Pochodzalla and co-workers, 1986c). The correlation functions exhibit a pronounced minimum at small relative momenta whlch becomes more pronounced with increasing energy of the emitted particles. The dotted curve shows finalstate model calculations (Chitwood and co-workers, ;985) based on deuteron-deuteron phase shifts extracted by the resonating group method (Chwieroth and co-workers, 1972). These phase

N u c l e u s - N u c l e u s Collisions

87

2.0

19JAu(14N, pp), E / A : 3 5 MeV I t

+ 1.5

A

= 35 °

,~ El+E2: ¢ 7;a-~OOMeV . o 5 0 - 75 MeV ~ • 2 4 - 50 MeV ,~ ~\~ r0 . . . . 3.5 fm ~1

(~4t

...... 5 f m

1.0

O Fig.

*÷ 20

40

60 80 q (MeV/c)

I00

120

42. Two-proton c o r r e l a t i o n f u n c t i o n s measured a t ® v135 o f o r *~N i n d u c e d r e a c t i o n s on 1'~Au at E / A - 3 ~ MeV. The constraints on the sum energy, El+E2, are indicated in the figure. (Poehodzalla and co-workers, 1 9 8 6 a ) 197Au(t4N,ad)X, E/A = 35 MeV

usu-85-sse

150-220 MeY

5

O0

lO0

200

0

100

200

0

0

100

100

200

q (MeV/c) Fig.

43. a-d correlation functions measured for *~N i n d u c e d reactions o n * ' ~ A u a t E / A = 3 5 MeY. D e t e c t i o n angles and energy constraints are given in the figure. ( C h l t w o o d and c o - w o r k e r s , 1986b)

shifts exhibit a resonant behaviour i n t h e £-1 p a r t i a l wave a t e n e r g i e s b e l o w 2 MeV. The predicted maximum of the correlation function at low relative momenta Is, however, not observed experlmentaliy. In fact, the experimental data can be reproduced rather nicely by calculations (Chltwood and co-workers, 1985) based on a more recent set of phase shifts extracted by the coupled channel R-matrix method (Hale and Dodder, 1984). The solid, dashed and dot-dashed curves show the results of these ealcuiatlons for sources of Gaussian spatial density and negligible lifetimes. Radius parameters extracted from two-deuteron correlatlon functions are considerably larger that those extracted from two-proton or a-d correlation functions (Chitwood

Claus-Konrad Gelbke and David H. Boal

88

and co-workers, 1985; Pochodzalla and co-workers, 1986a, 1986e). Such differences could indicate that correlations between different light particle pairs are sensitive to different stages of the reaction (Chitwood and co-workers, 1985). Source radii extracted from correlations between different particle pairs were compared by Pochodzalla and co-workers (1986a, 1986c). Table 2 gives the source parameters extracted for the ~*N+~'~Au reaction at E/A=35 MeV (Pochodzalla and co-workers, 1986a). Different source 2.s

. . . .

I

. . . .

i

~

1.5

ii / ~

T

1.0 -

/

0.5

I

group gfm] ?fm ~R-matriz

1

.... - -

". O

.

~

°

mo

L~y~" I//r 0

I ....

50

~

MeV o

0.0 [A" ,",,

. . . .

........ 8 f m resonating

ii _

I

E/A=BOMeV,e.v=30"t

I°VAu(~Ar,dd)X, 2.0

. . . .

vs-1

i

s

mt25-175UeV

]

I ....

!

100

I ....

150

200

q (UeV/o) Fig. 44. Two-deuteron correlation functions measured for ~°Ar induced reactions on *9~Au at E/A=60 MeV at Bay=30 °. The constraints on the sum energy, E~+E2, are given in the figure. (Pochodzalla and co-workers, 1986c)

Table 2.

Source radii, to, for a source of n~glL~Ible lifetime and Gausslan density, p(r)=po.exp(-r~/r~), extracted from two-partlcle correlation functions; the corresponding rms radii

are

given by: rrms = 43-~.r0;

equivalent

sharp

sphere radii are given by: R= 5 ~ . r o. The errors include normalization uncertainties for the different energy gates. The reaction studied is I*N+~S~Au at E/A-35 MeV. correlation p+p

d+d

EI+E 2 [MeV]

ro(35°) [fm]

24 50 50 - 75 75 - I00

4.9±0.5 4.3±0.3 3.8±0.2

30 80

8±2 5.5±1

-

-

-

80 160

t+t

36 - 120 120 - 200

p+a

52 - 1 0 0 100 - 150 150 - 200

6±0.5 5±0.5 4±0.5

d+ce

100 55 100 150 150 - 220

3.8±0.2 3.0±0.2 3.0±0.2

-

-

ro(50°) [fm] 5.2±0.5 4.0±0.3 3.6±0.2

6.5±I 5.5±I 6±0.5 4.8±0.3 3.7±0.3

3.9±0.2 2.8±0.2 2.7±0.2

89

Nucleus - Nucleus Collisions

p a r a m e t e r s were e x t r a c t e d for d i f f e r e n t p a r t i c l e pairs. The source sizes extracted for the emission of energetic p-p and e - d pairs are s m a l l e r than the slze of the t a r g e t nucleus, c o n s i s t e n t w l t h emission from regions of hlgh excitation which are smaller than the composite system. For the other particle pairs, larger s o u r c e d i m e n s i o n s are obtained. The e x t r a c t e d s o u r c e radii may be o r d e r e d approximately as follows: r0(a+d) ~ to(p+ p) $ to(m+ p) ~ r0(t+t), r0(d+d). Source parameters e x t r a c t e d for p a r t i c l e s of lower e n e r g i e s are larger, p o s s i b l y indicating emission from spatially expanding sources or emission at larger tlme intervals. The correlations measured for the ~ N + ~ 9 ~ A u r e a c t i o n are c o m p a r a b l e in m a g n i t u d e to those m e a s u r e d for ~sO i n d u c e d r e a c t i o n s on ~'~Au at E/A-25 MeV (Lynch and co-workers, 1983), but smaller than those measured for ~°Ar induced reactions on ~9~Au at E/A-60 MeV (Pochodzalla and c o - w o r k e r s , 1986c). At present, the detailed interpretation of these experimental results Is uncertain. There are several different effects which could account for the a p p a r e n t e j e c t l l e e n e r g y d e p e n d e n c e of the s o u r c e d i m e n s i o n s estimated In the zero lifetime limit. At the one extreme, one could take the m e a s u r e d s o u r c e d i m e n s i o n s at face value and c o n c l u d e t h a t energetic partlcles originate from a small volume at the early stage of the reaction; particles of lower energies would be emitted at slightly later stages at which more of the nucleons have i n t e r a c t e d ( r i g o r o u s l y speaking, thls i n t e r p r e t a t i o n already v i o l a t e s the assumption of negligible lifetime effects). An alternative interpretation was forwarded by Pratt (1984): for an e x p a n d i n g s y s t e m the c o l l e c t i v e flow velocity will introduce an energy dependence of the apparent source dimension. This effect can be expected to be i m p o r t a n t for high energy, h l g h m u l t i p l i c i t y r e a c t i o n s in which a large hot region is generated. Using a simplified model for the expanding region, Pratt (1984) was able to reproduce the ejectile energy dependence of twoplon correlations measured for Ar+KCI collisions at E/A=I.5 GeV. The importance of this effect at lower energies Is less clear.

NUMER~AL SIMU~TION$ • 1.0

EI+E 2 • 80 MeV

o~
10

20

T|

~

"

40

~p (MeV/c) Fla. 45. C o m p u t e r s i m u l a t i o n of two proton correlation functions predicted f o r an i s o l a t e d n u c l e u s at a n i n i t i a l temperature of 15 M e V . The s m o o t h curves c o r r e s p o n d p r e d i c t i o n s for G a u s s l a n sources of zero l i f e t i m e and radius parameters of r0= 4, 6, and 10 fm. (Boal and DeGulse, 1986) It is also p o s s i b l e that changes of the apparent source dimensions are caused by a dependence of the characteristic emission times on the k i n e t i c e n e r g y of the e m i t t e d p a r t i c l e s . Thls e f f e c t s h o u l d be i m p o r t a n t at l o w e r e n e r g i e s for w h i c h the emission times associated wlth equilibrated systems may be fairly long. To evaluate the m a g n i t u d e of thls effect, Boal and D e G u l s e (1986) p e r f o r m e d n u m e r i c a l simulations of the correlation function expected for the decay of a single nucleus at a f i n i t e t e m p e r a t u r e . In order to a c h i e v e s u f f i c i e n t l y r a p i d computer execution times and obtain calculations of sufficient statistical accuracy the nuclear potential was approximated ~ a f i x e d s t e p f u n c t i o n p o t e n t i a l . The n u c l e o n s i n t e r a c t e d by s c a t t e r i n g f r o m each other wlth their strong interaction cross section; the Coulomb repulsion between p r o t o ~ was included. As expected, the c a l c u l a t l o ~ predicted that energetic p a r t i c l e s

90

Claus-Konrad Gelbke and David H. Boal

left the system rather rapidly. As the residual system cooled by particle emission, more particles were emitted with lower average kinetic energies and at larger a v e r a g e time intervals. Figure 45 shows the calculated two-proton correlation functions for several constraints on the summed kinetic energies. The smooth curves show the correlation functions calculated in the zero lifetime approximation from the Coulomb interaction between the outgoing particles. Such an analysis of the computer generated data would yield s m a l l e r s o u r c e dimensions for more energetic particles. However, in the simulation the source dimension was kept fixed. The variation of the correlation function is entirely due to the fact that the average emission time depends on the energy of the emitted particle. Several issues need further investigation: At present, little is known about the impact parameter dependence of two-particle correlation functions. Different reaction products may have different impact parameter weightings (Beauvals and co-workers, 1986), see for example Flg. 32. Furthermore, different particle species may go out of equilibrium at different densities, depending on their interaction cross sections (Boal and Shlllcock, 1986). Sequential feeding from highly excited primary reaction products could also alter two-particle correlation functions. Finally It should be realized that calculations of theoretical correlation functions involve several uncertainties. The effects of long range Coulomb interactions wlth the residual nuclear matter have not yet been treated reliably. In addition, there are still uncertainties In the low energy phase shifts, particularly for the t+t system (Pochodzalla and co-workers, 1986a). First experimental hints concerning the relevance of some of these questions will be discussed below. Up to now, the sensitivity of two-particle correlations to the space-tlme characteristics of the emitting system has only been derived in terms of the thermal model (Jennlngs and coworkers, 1986) or the final state interaction model (Koonln, 1977). For sufficiently short emission time scales, the two models are essentially equivalent (Jennings and co-workers, 1986). However, for the case of particle emission from fully equilibrated nuclei, the temporal sequence of individual emission processes cannot be expected to be negligible. In qualitative terms, such decays are expected to lead to reduced correlations due to the longer lifetimes of the compound nuclei. For such decay processes, Koonln's formulation may be less useful for the calculation o£ two-particle correlation functions and alternative statistical formulations, such as appropriate generalizations of the Hauser-Feshbach theory, can be explored. The first such calculations were performed by Bernstein and co-workers (1984, 1985). I

I

I

I

|

I

O)160+27AI -~ 2 p + X E / A -" 25 MeV 1.5

I

0.5

,,

b) leO+ 12C-P 2p +X E / A = 25 MeV

O.C

Fig.

O

~

,

,

,

J I

~

50 Ap (MeV/c)

,

,

,/

IOO

46. T w o - p r o t o n correlation functions measured for ~'0 induced reactions on *ZC and 2"A1 a t E / A - 2 5 MeV a r e c o m p a r e d t o calculations within the framework of the statistical model for the decay of equilibrated compound n u c l e i . (Bernstein

and co-workers, 1985)

Nucleus

-

91

Nucleus Collisions

Figure 46 shows two-proton correlation functions measured for ~*0 induced reactions on ~2C and 2~AI at E/A-25 MeV (Bernstein and co-workers; 1985). For these reactions, the inclusive light charged particle spectra are compatible with considerable contributions from statistically emitting compound-like residues - in contrast to reactions on ~9~Au for which nonoompound emission processes dominate the cross sections (Tsang and co-workers, 19840). It is noteworthy that the two-proton correlations measured for ~60 induced reactions on ~2C and 27AI are smaller than those measured for reactions on le~Au (Lynch and co-workers, 1983). Within Koonin's model, these differences can be qualitatively understood as arising from differences in emission time scales for equilibrium and non-equilibrlum processes. Quantitative calculations along these lines of argument are, however, not yet available. The solid lines in Fig. 46 correspond to Hauser-Feshbach calculations for emission from a fully equilibrated compound nucleus (Bernsteln and co-workers, 1985). In these calculations, coincident protons were assumed to be emitted either sequentially, with negligible final state interactions, or in the form of particle unstable "2He nuclei" which decay after emission. The measured correlation functions are quantitatively reproduced by these calculations. Unfortunately, the theory has not yet been extended to include particle emission during the early non-equilibrated stages of the reaction or correlations resulting from non-resonant interactions. Consistent statistical treatments of the temporal evolution of the reaction remain a challenging problem for future research.

8.2. Coulomb distortions In our previous discussion of two-particle correlations, we have neglected modifications of the wave functions of relative motion of the two detected particles by the Coulomb interaction with the remaining nuclear system. Such Coulomb distortions are expected to be less important for the case of narrow resonances or for decays into two particles with identical charge-to-mass ratios. They are, however, not negligible for the case of short-llved nuclear resonances which decay into two light particles of different charge-to-mass ratio.

Tc.m.(MeV) 0.5 1.0 2.0

0

t

(9B)

i

I

5 i

I

IO

I

I

5Li

1.5

i

15

I I I I I I I

o

v >v=

.

v,,v, g

+

n.-

....

1.0 '~

0.5

,,,

197A,, ( 40A

~ ~,

E(A" 60 MeV' el°V"30° 50 100

-~

7 150

q (MeV/c) Fig. 47. Correlation function for coincident protons and alpha particles for ~°Ar induced reactions on 197Au at E/A-60 MeV. Open and solid points correspond to the constraints v < v and v > v respectively. (Pochodzalla and coa~rke?s, 1985%) P' An example of line-shape distortions due to final-state interactions with the Coulomb field of the residual nuclear system is given in Fig. 47 (Pochodzalla and co-workers, 1985b). The figure shows p-m correlation functions measured for ~°Ar induced reactions on *e?Au at E/A-60 MeV. For these correlation functions, the summation in Eq. 40 was performed with the constraints v < v (open points) and v > v (solid points), where v and v denote the laboratory velocities o~ alpha particles a~d p~otons, respectively. TheUexperi~ental correlation functions exhibit two pronounced maxima located at q - 15 and 50 MeV/c. While the location of the first maximum (q-15

PPHP-G

92

Claus-Konrad G¢lbke and David H. Boal

MeV/c) is the same for both kinematic branches, the location of the second maximum (q-50 MeV/c) is not. The peak near q-15 MeV/c is due to the detection of two of the three fragments from the two stage decay of 9B, 'B ~ p+'Be~ p+(s+s); it is a result of the small decay energies and the narrow widths of the g r o u n d states of 9B and 'Be. B e c a u s e of the long l i f e t i m e s of these states, d i s t o r t i o n s in the Coulomb field of the residual nucleus are negligible (Pochodzalla and co-workers, 1985b). The peak near q-50 MeV/c is related to the unbound ground state of 5Li. This state has a width of about 1.5 MeV corresponding to a short mean life of about %=130 fm/c. Since the charge-to-mass ratio of protons is greater than that of alpha p a r t i c l e s the former will experience greater accelerations in the Coulomb field of the residual nuclear system. As a consequence, the a s y m p t o t i c v e l o c i t y d i f f e r e n c e b e t w e e n p r o t o n s and a l p h a p a r t i c l e s will be decreased if v < v at the time of decay and increased if v > v . The solid lines in Fig. 47 show the results 8f s~mple classical calculations for this effect ~Pochodzalla and c o - w o r k e r s , 1985b). The agreement with the experimental correlation function is good.

,

laVAu(1~N,pd)X, E/A = 35 MeV 1.5

.........

. . . . . . . . .

,

. . . . . . . . .

,

. . . . . . . . . . . . . . . . . . .

30 M e V

1.0

~ . . . . . . . .

,

. . . . . . . . .

,

. . . . . . . . .

@~v=50°

i® a v = 3 5 ° _~ T -~ 50 M e V

÷"*'' i i

o.s

30

MeV S T

50 MeV

50

M e V _~ T

70

'';

il i i : r "~

50 MeV s

T < 70

MeV

MeV

~ '~ -'0 +~ 5 01 I (,**,N ~ % ~'.~/,,,,, , 70 M e V

/

-~

< 100 M e V

i 70 MeV - T < 100 MeVi '

++

: i i J

0.5 i i

Ue=O

0.0

........

-20

] .........

-10

J Uc=O

i i S¢ I.........

0

i ~ S©

l .........

10

20 - 1 0 S (MeV)

0

10

20

Fig. 48. Correlation function for coincident protons and d e u t e r o n s as a f u n c t i o n of the c o o r d i n a t e S, Eq. 54, measured for the *~N+*9~Au reaction at E/A=35 MeV. The s u m m a t i o n s in Eq. 55 were p e r f o r m e d over d e t e c t o r pairs with angular separations of A8-6.1 °. E n e r g y t h r e s h o l d s E ~12 MeV and Ed~15 MeV were applied to the summation in addition to the constraints on T indicated in the figure. The values, SO, c o r r e s p o n d i n g to m i n i m u m r e l a t i v e velocity (v_-v~) are m a r k e d by d o t t e d l i n e s ; d a s h e d l i n e s i n d i ~ a t ~ t h e locations of the measured minima of R(S). (Pochodzalla and co-workers, 1986b) In the a b s e n c e of l o n g - l i v e d final states, correlations between two particles with different mass-to-charge ratios are sensitive to the Coulomb field of the residual nuclear system at the p o i n t of emission. The s t r e n g t h of the C o u l o m b f i e l d of the residual system at this point d e p e n d s on the t e m p o r a l e v o l u t i o n of the r e a c t i o n and can s e r v e to a s s e s s the r e l a t i v e i m p o r t a n c e of s e q u e n t i a l e m i s s i o n p r o c e s s e s . (Sequential emission processes occur at large distances from the residual nuclear system where the Coulomb field is small.) P r o t o n s and d e u t e r o n s have different charge-to-mass ratios and experlenee non-resonant final state interactions. When Coulomb distortions by the residual nuclear system are negligible, the Coulomb repulsion between these two particles will produce a minimum at zero relative velocity.

Nucleus

-

Nucleus Collisions

93

The position of this minimum can be shifted by Coulomb distortions in the field of the residual n u c l e a r s y s t e m ( P o c h o d z a l l a and c o - w o r k e r s , 1986b). In order to demonstrate this effect, Pochodzalla and co-workers (1986b) defined new variables, T and S, in terms of the p r o t o n and deuteron energies, Ep and Ed: S = (~-Ep- Ed)/WT~-~-~ ,

T = (Ep÷ ~ . E d ) / ~

,

(54)

where s =md/m p , and defined the correlation function R(S): T2 T2 f dT-Ypd(S,T) = C.(I+H(S))-~ I dT'Yp[Ep(S,T)]'Yd[Ed(S,T)] • Tl Tl

(55)

These variables correspond to coordinates measured along axes parallel and perpendicular to the line Vd-V which corresponds to S=O. Coulomb distortions are manifest by d i s p l a c e m e n t s of the minima of PR(s) from the location S=O. Figure 48 shows proton-deuteron correlation functions, R(S), measured for ~ N induced reactions on ~9~Au at E / A = 3 5 MeV at e m i s s i o n angles of 0 =35 ° and 50 ° . The minima of the correlation functions are clearly displaced from the l o c a t i o ~ V S = 0 . P o c h o d z a l l a and c o - w o r k e r s (1986b) a n a l y z e d the m a g n i t u d e of this d i s p l a c e m e n t by p e r f o r m i n g c l a s s i c a l t h r e e - b o d y C o u l o m b trajectory calculations treating the mutual Coulomb interactions b e t w e e n the d e t e c t e d p r o t o n and deuteron, and the (undetected) composite nuclear system. In terms of these calculations, the observed location of the minimum of R(S) indicates a distance of emission f r o m the center of the c o m p o s i t e n u c l e u s of d < 15 fm, c o n s i s t e n t w i t h emission from a point close to the surface of the composite system. This short distance excludes significant contributions to the y i e l d of c o i n c i d e n t protons and deuterons from the sequential decay of long-llved projectile fragments. It is not anticipated that more exact c a l c u l a t i o n s will m o d i f y this q u a l i t a t i v e conclusion; however, more consistent treatments of the nuclear and Coulomb interactions between the emitted particles can be expected to place more quantitative c o n s t r a i n t s on the e m i t t i n g system.

8.3. Reaction filters W h i l e two-particle inclusive measurements can provide insights about the average properties of the e m i t t i n g system, m o r e d e t a i l e d i n f o r m a t i o n m u s t be o b t a i n e d f r o m m o r e e x c l u s i v e measurements in which specific classes of reactions can be suppressed or enhanced. One can, for example, discriminate between quasi-elastic and more violent, f u s i o n - l i k e p r o j e c t i l e - t a r g e t interactions by m e a s u r i n g the l i n e a r m o m e n t u m t r a n s f e r to the h e a v y r e a c t i o n residue. Alternatively, one can hope to classify the violence of the p r o j e c t i l e target i n t e r a c t i o n in t e r m s of the light p a r t i c l e m u l t i p l i c i t y ; this technique is most useful at higher energies where the light particle multiplicity may be related to the number of p a r t i c i p a n t n u c l e o n s in the region of geometrical overlap between projectile and target. Information about the linear momentum transfer can be extracted from the f o l d i n g angle, elf, b e t w e e n two coincident fission fragments, see also Fig. 2. Measurements of two light particles in coincidence with two fission fragments were performed by Chen and co-workers (1986) for I~N i n d u c e d r e a c t i o n s on ~9~Au at E/A=35 MeV. The left hand part of Fig. 49 shows folding angle distributions measured inclusively (open points) and in coincidence with e-d p a i r s e m i t t e d at e av ,20 °. The upper scale indicates the average linear momentum transfer to the heavy reaction residue, Ap/p, measured in units of the p r o j e c t i l e m o m e n t u m , p. The i n c l u s i v e d i s t r i b u t i o n e x h i b i t s a m a x i m u m at A p / p - 0 . 8 , i n d i c a t i n g d o m i n a n t c o n t r i b u t i o n s from incomplete fusion reactions. Small folding angles are suppressed when c o i n c i d e n t s-d pairs are detected; this suppression is largely due to momentum conservation (coincident e-d pairs carry away an average linear momentum of 0.35p). In order to explore the dependence of light particle energy s p e c t r a and correlation functions on the linear momentum transferred to the heavy target residue, three gates on the folding angle were defined as shown in the figure. The right hand side of Fig. 49 s h o w s the d e p e n d e n c e of the a-d coincidence yield on the total kinetic energy, Ed+E , for ~-d p a i r s w h i c h can be a t t r i b u t e d to the 2.186 MeV state of 6Li. For small l i n e a r ~ m o m e n t u m t r a n s f e r s (gate 3) the ~-d coincidence yield is dominated by a strong quasi-elastic component which is centered close to the projectile velocity. This component is s t r o n g l y s u p p r e s s e d by g a t i n g on large l i n e a r m o m e n t u m transfers (gate I). These observations support the intuitive picture which associates small linear m o m e n t u m t r a n s f e r s w i t h c o l l i s i o n s at l a r g e i m p a c t parameters for which break-up reactions or sequential decays of excited projectile residues are important. Large linear momentum transfers are associated with less peripheral and more violent c o l l i s i o n s in w h i c h the major part of the projectile momentum is absorbed by the heavy target nucleus.

94

Claus-Konrad Gelbke and David H. Boal

Ap/p i04

"'I .......... ~4N+I~Au, E/A=35MeV, '

I .... I .... I @av=20 °

'

10 s oI qU'l

o o

~i

o

108

(D~ o

•

-

o o

o o

•

o

•

%i%d+a

: • ',®

T a . = . = 0 . 3 - 1 . 2 MeV' .... I .... I .... I

10 lj

100

150

o . (d g)

0

i00

~.00

300

E d + E a (MeV)

Fig. 49. Left hand part: Folding angle distributions measured inclusively (open points) and in coincidence with a-d pairs (full p o i n t s ) . The d a s h e d lines indicate the boundaries for gates I, 2 and 3. R i g h t hand part: D e p e n d e n c e of the u-d coincidence yield on the total kinetic e n e r g y , E ~ 4 E , for e-d pairs w h i c h can be attributed to the 2~;8~MeV state in 6Li. The spectra are gated as indicated. The vertical scale is in arbitrary units. (Chen and co-workers, 1986) Figure 50 shows the s-d correlation function measured for large (gate I, open points) and small (gate 3, solid points) linear momentum transfers to the heavy reaction residue. In order to reduce contributions from later stages of the reaction for which sequential emission of low e n e r g y p a r t i c l e s from the fully equilibrated composite system may be important, these correlations were gated on total energies well above the compound nucleus Coulomb barrier. The selection of events with large linear momentum transfer produces enhanced correlations. For orientation, the curves in Fig. 50 show correlation functions predicted by the final-state interaction model (Koonin, 1977; Boal and Shillcock, 1986) for sources of Gaussian density and negligible lifetime. For gates I and 3, source radii of r0-2.8 and 3.6 fm are extracted from the sharp peak of the s-d correlation function at q-42 MeV/c, respectively. (Source radii extracted from the broad peak at q-85 MeV/c are larger by about 0.8 fm; the reason for thls d i f f e r e n c e is not u n d e r s t o o d at present.) Consistent wlth the results from inclusive measurements, these source dimensions are smaller than the size of the t a r g e t n u c l e u s (r0(Au)-4.4 fm}. However, a strictly geometrlc interpretation of the correlation functions would imply that quasi-elastic collisions are characterized by sources significantly larger than the size of the projectile nucleus [r0(N)-2.1 fm); it would, furthermore, imply that quasi-elastlc collisions produce sources of larger dimensions than fuslon-llke collisions. Such implications are difficult to understand. It is more llkeiy that temporal effects cannot be neglected when Interpreting the correlation functions. The observation of reduced oorrelatlons for peripheral processes could, therefore, reflect longer emission time scales rather than larger source dimensions. For example, sequential decays of excited and fully equilibrated projectile residues could involve longer emission time scales than preequillbrium reactions. Significant contributions from the decay of fully equilibrated nuclei are expected to result In reduced correlation. For fuslon-llke collisions, contributions from the decay of equilibrated projectile residues are suppressed. It is significant that these collisions exhibit larger correlations. They should reflect, more clearly, the spatial localization of the reaction (Chen and co-workers, 1986).

Nucleus - Nucleus Collisions

i~VAu(14N,da if), E / A = 3 5 M e V ,

95

eav=20 °

E~+Ed~ I00 MeV

20 2

o eff= 120°- 145 ° • e f f = 1 6 0 ° - 180° ....... 2.5fm A - - 3.fro 1 ~ t l , ! ~ ~ - - - 4.fm I

+

50

I

100 q (MeV/c)

150

Fig. 50. a-d correlation functions gated on large (gate I, open points) and small (gate 3, solid points) linear momentum transfers to the heavy reaction residue. The gate on the total kinetic energy is indicated in the figure; the curves are explained in the text. (Chert and co-workers, 1986) Figure 51 summarizes the present information about the dependences of ~-d correlation functions on the total energy per nucleon, (EIA)~o~-(E +E~)I6, of the two coincident particles, on emission angle and on reaction type ~ l~+1~Au collisions at EIA-35 MeV (Chen and coworkers, 1986). Inclusive correlations and correlations measured in coincidence with fission fragments ape shown in the right and left hand parts of the figure, respectively. The left hand scale of the figure gives the integral correlation, Heir- Idq-R(q), with the integration range of q-30-60 MeV/c; this quantity depends less on the resolution of the experimental apparatus than the maximum value of the correlation function. The right hand scale of the figure gives source radii, Pc, extracted in the limit of negligible lifetime. At low energies, the a-d correlations are of comparable magnitude. Fop small linear momentum transfers (gate 3), they ape nearly independent of energy. For larger linear momentum transfers, on the other hand, the a-d correlations depend strongly on energy. For inclusive correlations, shown in the right hand part of the figure, a similar energy dependence exists. It becomes more pronounced at larger emission angles where contributions from quasl-elastlc processes become less important thus lending support to the previous conclusion that contributions from the decay of equilibrated projectile residues lead to reduced two-particle correlations. The dependence of two-particle correlation functions on the multiplicity of the emitted light particles has not yet been measured for intermediate energy nucleus-nucleus collisions. A first step into this direction has, however, been undertaken by Kyanoweki and co-workers (1986) for ~°AP induced reactions on :9?Au at EIA-60 MeV. They classified the reactions in terms of the number, v, of charged particles detected in a forward angle scintillator array covering an angular range of about O-5o-30 °. The magnitude of the s-d correlation function was observed to depend dramatically upon the gating condition. Fop ~-0, the magnitude of the sharp peak in the correlation function is larger by about a factor of 3-4 as compared to the magnitude for v~10, see Fig. 52. The authors caution that these differences are likely to represent differences in emission times rather than spatial dimensions of the e m i t t i n g s o u r c e s . A l t h o u g h the interpretation of the number, ~ , of charged particles emitted into the forward direction is rather model dependent, the observed dramatic dependences of correlation functions on various

96

Claus-Konrad Gelbke and David H. Boal

14N+I~VAu

300

....

~ ....

E/A , ....

=

l .........

d-a-ff ear=20°

35MeV

l ....

J ....

+

÷ o

3

0

¢

cD

-- 2.5

÷

¢

200

>

, ....

d-~

+

+

w-4

3.5 v

v

o

I00

4 [120"-145" {~ff= ~ 1 4 5 ° - 1 6 0 t. 1 6 0 . - 1 8 0 ,,,I

0 0

10

....

I ....

20

• * 0 • •

8a,,= ....

I ....

30

0

50"

•

20 °

•

35"

I ....

10

I ....

20

I '5

m

6 I ....

30

40

(E/A)tot (MeV) Fig. 51. Dependence of a-d correlations on the total energy per nucleon of the outgoing particles. Correlations measured inclusively and in coincidence with fission fragments are shown in the right and left hand parts, respectively. The left hand scale corresponds to R ff- fdq.R(q), with the integration performed over the rang~ of q=30-60 MeV. The right hand scale gives source radii, ro, extracted for Gausslan sources of negligible lifetime. (Chen and coworkers, 1986) gating conditions should stimulate further, more detailed investigatlons of specific classes of interactions selected by appropriate reaction filters. The development and utilization of such reaction filters present a challenge to the skill and resourcefulness of experimental and theoretical nuclear physicists alike. 9. CONCLUSION Intermediate energy nucleus-nucleus collisions are sufficiently violent to render pure mean field descriptions inadequate. At the same time, the energies are not high enough to warrant the neglect of the mean nuclear field. Realistic microscopic models must be able to treat the intricate interplay between nuclear binding and individual nucleon-nucleon collisions which in turn are strongly affected by the Pauli principle. A considerable number of experiments have been performed which delineate the systematic trends of incomplete fusion processes and particle emission from the early, nonequillbrated stages of the reaction. Significant progress has been made towards a microscopic understanding of these data. Numerical solutions of kinetic equations incorporating individual nucleon-nucleon collisions, Pauli blocking and the mean nuclear fleid reproduce the qualitative trends of linear momentum transfer in fuslon-like reactions, the apparent thermalizatlon of the kinetic energy spectra of particles emitted at intermediate rapldlties, the preferential emission of particles in the reaction plane, and their slightly enhanced emission to negative deflection angles. However, most calculations are limited to the description of single nucleon densities. Detailed comparisons with the results from more restrictive coincidence experiments are hampered by the neglect of fluctuations and correlations which are necessary ingredients for the treatment of complex particle emission. Treatments of complex particle emission are often based on statistical approaches many of which use the simplifying assumption of local equilibrium. Chemical emission temperatures extracted from the relative populations of states are, in general, lower than kinetic temperatures which characterize the kinetic energy spectra of the emitted particles. Thls qualitative behaviour Is expected for adiabatically expanded systems. Although the quantitative relation between

Nucleus

I

I

I

-

Nucleus Collisions

I

l

I

l

I

I

97

I

;OAr +lSTAu-- d+= ~' E/A = 60 MeV

12 I0

*,=0

8 o- 6

n-. +

t

-4

s, 2 0

, , , I I I ', I ', I

A

4

='=I0

, %

2 I

0 0

20

40

I

I

I

60

I

80

I

I

I

I00

q (MeV/c) Fig. 52. a - d c o r r e l a t i o n f u n c t i o n s measured for ~°Ar induced reactions on *'~Au at E/A=60 MeV. The gating condition c o r r e s p o n d i n g to the number, ~, of charged particles detected in a forward angle scintillator array is given in the figure. (Kyanowski and co-workers, 1986) chemical and kinetic temperatures is not yet understood on a microscopic level, there is encouraging qualitative agreement with recent microscopic calculations which incorporate correlations and fluctuations on a semiclasslcal level. The overall importance of complex particle emission in high-lying particle unstable states and the ensuing secondary decay processes can lead to characteristic structures in the mass d i s t r i b u t i o n s of intermediate mass fragments. Secondary decays also alter the relative populations of states, leading to complications in the interpretation of chemical emission temperatures. The detailed understanding of secondary decay processes is a prerequisite for attempts to extract information about the liquld-gas phase transition of nuclear matter. These difficulties, as well as the lack of realistic reaction calculations, reflect in part the ambiguities which still persist for the interpretation of complex fragment emission and its relation to the phase transition. At the time of writing, several promising theories are being developed which may bring significant advances to our understanding of complex particle emission processes. The concept of spatial localization derives experimental support from measurements of twoparticle correlation functions at small relative momenta. The full information content of such measurements has not been exploited yet. In more qualitative terms, these measurements give strong indications about the importance of temporal effects: the assumption of emission from a static subsystem is in conflict with the measured energy dependence of two-particle correlation functions in fuslon-like collisions. The strong sensitivity of two-particle correlations to specific reaction filters indicates that they may provide a useful tool for probing the spacetime evolution of specific classes of reactions. There is hope that new numerical simulation techniques will bring further progress in exploiting two-particle correlation functions as a diagnostic tool of the reaction dynamics.

ACKNOWLEDGEMENTs This work was supported in part by the National Science Foundation under Grant No. PHY 83-12245 and by the Natural Sciences and Engineering Research Council of Canada.

98

Claus-Konrad Oelbke and David H. Boal REFERENCES

Aiehelln, J. (1984). Phys. Rev. Lett., 522, 2340. Aichelin, J., and O. Bertsch (1984). Phys. Lett., 138B, 350. Aichelln, J., and J. HUfner (1984). Phys. Lett., 13-6~, 15. Aichelln, J., J. HBfner and R. Ibarra (1984). Phys. Rev., C30, 107. Aichelin, J. (1985). Phys. Lett., 164B, 261. Aichelin, J., and G. Bertsch (1985). Phys. Rev., C31, 1730. Aichelin, J., and H. StOcker (1985). Phys. Lett.,'-~3B, 59. Aichelin, J., and H. St~eker (1986). Phys. Lett., (in press). Alamanos, N., P. Braun-Munzlnger, R. F. Frelfelder, P. Paul, J. Stachel, T. C. Awes, R. L. Ferguson, F. E. Obenshain, F. Plasil, and G. R. Young (1986). Phys. Lett. 173B, 392. Ardouln, D., H. Delagrange, H. Doubre, C. Gr~goire, A. Kyanowski, W. Mittlg, A. P~ghaire, J. P~ter, F. Salnt-Laurent, Y.P. Viyogi, B. Zwieglinski, J. Pochodzalla, C.K. Gelbke, W. Lynch, M. Maler, G. Bizard, F. Lef~bvres, B. Tamaln, and J. Qu~bert (1985). Nucl. Phys., A447, 585C. Arnell, S. E., S. Mattsson, H. A. Roth, O. Skeppstedt, S. A. Hjorth, A. Johnson, A. Kerek, J. Nyberg, and L. Westerberg (1983). Phys. Lett., 129B, 23. Auble, R. L., J. B. Ball, F. E. Bertrand, R. L. Ferguson, C. B. Fulmer, I. Y. Lee, R. L. Robinson, G. R. Young, J. R. Wu, J. C. Wells, and H. Yamada (1982a). Phys. Bey., C25, 2504. Auble, R. L., J. B. Bail, F. E. Bertrand, C. B. Fulmer, D. C. Hensley, I. Y. Lee, R. L. Robinson, P. H. Stelson, D. L. Hendrle, H. D. Holmgren, J. D. Silk, and H. Breuer (1982b). Phys. Rev. Lett., 49, 441. Auble, R. L., J. B. Ball, F. E. Bertrand, C. B. Fulmer, D. C. Hensley, I. Y. Lee, R. L. Robinson, P. H. Stelson, C. Y. Wong, D. L. Hendrle, H. D. Holmgren, and J. D. Silk (1983). Phys. Rev., C28, 1552. Awes, T. C., C. K. Gelbke, B. B. Back, A. C. Mignerey, K. L. Wolf, P. Dyer, H. Breuer, and V. E. Viola, Jr., (1979). Phys. Lett., 87B, 43. Awes, T. C., G. Poggl, C. K. Gelbke, B. B. Back, B. G. Glagola, H. Breuer, and V. E. Viola, Jr., (1981a). Phys. Rev., C24, 89. Awes, T. C., G. Poggi, S. Salnl, C. K. Gelbke, R. Legraln, and G. D. Westfall (1981b). Phys. Lett., I03B, 417. Awes, T. C., S. Salni, G. Poggl, C. K. Gelbke, D. Cha, R. Legain, and G. D. Westfall (1982). Phys. Rev., C25, 2361. Back, B. B., and S. Bj~rnholm (1978). Nucl. Phys., A302, 343. Back, B. B., K. L. Wolf, A. C. Mignerey, C. K. Gelbke, T. C. Awes, H. Breuer, V. E. Viola, Jr., and P. Dyer (1980). Phys. Rev., C22, 1927. Ball, J. B., C.B. Fulmer, M.L. Mallory, and R.L. Robinson (1978). Phys. Rev. Lett., 40, 1698. Balescu, R. (1975). Equilibrium and Nonequilibrihm Statistical Mechanics, Wiley, New York. Ban-Hao, S., and D. H. E. Gross (1985). Nuel. Phys., A437, 643. Barker, J. H., J. R. Beene. M. L. Halbert, D. C. Hensley, M. JSaskelalnen, D. G. Sarantites, and R. Woodward (1980). Phys. Rev. Lett., 45, 424. Barz, H. W., J. P. Bondorf, R. Donangelo, and H. Sehulz (1986). Phys. Lett., !69B, 318. Bauer, W., D. R. Dean, U. Mosel, and U. Post (1985). Phzs. Lett., 150B, 53. Bauer, W., W. Cassing, U. Mosel, M. Toyama, and R. Y. cusson (1986a). Nucl. Phys., A456, 159. Bauer, W., G. F. Bertsch, W. Casslng, and U. Mosel (1986b). MichiGan State University Preprlnt. Bauer, W., U. Post, D. R. Dean, and U. Mosel (1986c). Nucl. Phys., A452, 699. Bauer, W., G. F. Bertsch, and S. Das Gupta (1986d). MichiGan State University Preprlnt. Beard, K. B., W. Benenson, C. Bloch, E. Kashy, J. Stevenson, D. J. Morrisey, J. van der Plicht, B. Sherrill, and J. S. Winfield (1985). Phys. Rev., C32, 1111. Beauvals, G. E., D. H. Boal, and J. C. K. Wong (1986). Phi?. Bey., ~ (in press). Beavis, D., S. Y. Fung, W. Gorn, A. Huie, D. Keane, J. J. Lu, R. T. Poe, B. C. Shen, and G. VanDahlen (1983a). Ph[s. Rev., C27, 910. Beavls, D., S. Y. Chu, S. Y. Fung, W. Gorn, D. Keane, R. T. Poe, G. VanDahlen (1983b). Phys. Rev., C28, 2561. Benenson, W., G. Bertsch, G. M. Crawley, E. Kashy, J. A. Nolen, J r . , , H . Bowman, J. G. I n g e r s o l l , J. O. Rasmussen, J. S u l l l v a n , M. Kolke, M. Sasao, J. Peter, and T. E. Ward (1979). Phys. Rev. L e t t . , 4_.33, 683. Bernsteln, M. A., W. A. Friedman, and W. G. Lynch (1984). Phys. Rev., C29, 132 and C30, 412(~). Bernsteln, M. A., W. A. Friedman, W. G. Lynch, C. B. Chltwood, D. J. F i e l d s , C. K. Gelbke, M. B. Tsang, T. C. Awes, R. L. Ferguson, F. E. Obenshaln, F. P l a s i l , R. L. Robinson, and G. R. Young (1985). Phys. Rev. L e t t . , 54, 402. B e r t l n l , H. W., R. T. Santoro, and O. W."Hermann (1976). Phys. Rev., CI.~4, 590. Bertsch, O., and D. Mundlnger (1978). Phys. Rev., C17, 1646. Bertsch, G. (1980a). In D. Wilkinson (Ed.), ProB. Part. Nucl. Phys., Vol. 4, Pergamon, Oxford, p. 483. Bertsch, O. (1980b). Nature, 283, 280.

Nucleus

-

Nucleus Collisions

99

Bertsch, G., and J. Cugnon (1981). Phys. Bey., C24, 2514. Bertsch, O., and P. J. Siemens (1983). Phxs. Lett., 126B, 9. Bertseh, G., H. Kruse, and S. Das Gupta (1984). Phys. Bey., C29, 6?3. Bhowmik, R. K., E. C. Pollaeoo, a. B. A. England, G. C. Morrison, and N. E. Sanderson (1981). Nuel. Phys., £363, 516. Bhowmlk, B. K., J. van Drlel, B. H. Siemssen, G. J. Balster, P. B. Goldhoorn, S. GonggriJp, Y. Iwasakl, B. V. F. aanssens, H. Sakai, K. Siwek-Wilezynska, W. A. Sterrenburg, and J. Wilczynskl (1982). Nuel. Phys., A390, 117. Birkelund, J. B., and J. B. Huizenga (--1383). Annu. Bey. Nucl. Sci., 33, 265. Biro, T. S., and a. Knoll (1985). Phys. Lett., 165B, 256. Biro, T. S., J. Knoll, and a. Richert (1985). Preprint, GS185-6?. Blann, M. (1985a). Phys. Bey., C31, 295. Blann, M. (1985b). Phys. Bey., C31, 1245. Bloeh, C., W. Benenson, E. Kashy, D. J. Morrissey, R. A. Blue, R. M. Ronnlngen, and H. Utsunomiya (1986). Phys. Rev., C3___44,850. Boal, D. H. (1983). Phys. Bey., C28, 2568. Boal, 0. H. (1984a). Phys. Bey., C30, 119. Boal, D. H. (1984b). Phys. Bey., C30, 749. Boal, D. H., and J. H. Reid (1984). Phys. Bey., C29, 973. Boal, D. H., and H. DeGuise (1986). Phys. Bey. Lett., 5~?, 2901. Boal, D. H., and A. L. Goodman (1986). Phys. Bev., C33, 1690. Boal, D. H., and a. C. Shillcock (1986), Phys. Rev., C33, 549. Bodmer, A. R., C. N. Panes, and A. D. MaeKellar (198054"-.Phys. Rev., C22, 1025. Bohrmann, S., J. HOfner, and M. C. Nemes (1983). Phys. Lett., 120B, 59. Bonche, P., S. Levlt, and D. Vautherin (1984). Nucl. Phys., A427, 278. Bonche, P., S. Levit, and D. Vautherin (1985). Nuol. Phys., A-'4~, 265. Bondorf, J. P., J. N. De, A. O. T. Karvinen, G. Fai, and B. Jakobsson (1979). Phys. Left., 84B, 162. Bondorf, J. P., J. N. De, G. Pal, A. O. T. Karvinen, B. Jakobsson, and J. Bandrup (1980). Nuel. Phys., A333, 285. Bondorf, R. Donangelo, I. N. Mishustin, C. Pethick, and K. Sneppen (1985a). Phys. Lett., [50B, 57. Bondorf, J. P., R. Donangelo, I. N. Mishustin, C. J. Pethick, H. Sehulz, and K. Sneppen (1985b). Nucl. Phys., A443, 321. Bordsrle, B., M. F. Rivet, I. Forest, J. Galln, D. Guerreau, R. Bimbot, D. Gardes, B. Gatty, M. Lefort, H. Oesehler, S. Song, B. Tamaln, and X. Tarrago (1983). Nucl. Phys., A402, 57. Borderle, B., M. F. Rivet, 0. Cabot, D. Fabrls, D. Gard~s, H. Gauvln, F. Hannappe, and J. P~ter (1984). z. Phys., A318, 315. Borderie, B., and M. F. Rivet (1985). Z. Phys., A312 703. Brosa, U., and W. Krone (1981). Phys. L e t t . , I05B, 22. B r i t t , H. C., and A. R. Quinton (1961). Phys. Rev., 124, 877. B u t l e r , S. T., and C. A. Pearson (1961). Phys. Rev. L e t t . , [ , 69. Butler, S. T., and C. A. Pearson (1962). Phys. Lett., I, 77. Campi, X., J. Desbois, and E. Lipparini (1984). Phys. Eett., 142B, 8. Caskey, G., A. Galonsky, B. Remington, M. B. Tsang, C. K. Gelbke, A. Kiss, F. Deak, Z. Seres, J. J. Kolata, J. Hinnefeld, and J. Kasagi (1985). Phys. Rev., C3_._!] , 1597. Chan, Y., M. Murphy, R. G. Stokstad, I. Tserruya, S. Wald, and A. Budzanowskl (1983). Phys. Rev., C27, 447. Chen, Z., C. K. Gelbke, J. Pochodzalla, C. B. Chitwood, D. J. Fields, W. G. Lynch, and M. B. Tsang (1986). Michigan State University Preprint MSUCL-567. Chiavasso, E., S. Costa, G. Dellaeasa, N. DeMarco, M. Gallio, A. Musso, E. Aslanldes, P. Fassnacht, F. Hibou, T. Bressanl, M. Carla, S. Serei, F. lazzi, and B. Mlnettl (1984). Nucl. Phys., A422, 621. Chitwood, C. B., D. J. Fields, C. K. Gelbke, W. G. Lynch, A. D. Panaglotou, M. B. Tsang, H. Utsunomlya, and W. A. Friedman (1983). Phys. Lstt., 131B, 289. Chitwood, C. B., J. Aichelin, D. H. Boal, G. Bertsch, D. J. Fields, C. K. Gelbke, W. G. Lynoh, M. B. Tsang, J. C. Shillcock, T. C. Awes, R. L. Ferguson, F. E. Obenshain, F. Plasil, R. L. Robinson, and G. R. Young (1985). Phys. Rev. Lett., 54, 302. Chitwood, C. B., D. J. Fields, C. K. Gelbke, D. R. Klesch, W. G. Lynch, M. B. Tsang, T. C. Awes, R. L. Ferguson, F. E. Obenshaln, F. Plasil, R. L. Robinson, and G. R. Xoung (1986a}. Phys. Rev., C34, 858. Chitwood, C. B., C. K. Gelbke, J. Poehodzalla, Z. Chen, D. J. FielOs, W. G. Lynch, B. Morse, M. B. Tsang, D. H. Boal, and J. C. Shillcock (1986b). Phys. Left., 172B, 27. Chwieroth. F. S., Y. C. Tang, and D. B. Thompson (1972). Nuel. Phys., A189, I. Clare, R. B., and D. Strottman (1985). Phys. Rep, 141, 177. Coeconi, G. (1974). Phys. Left., 49B, 459.

100

Claus-Konrad Gelbke and David H. Boal

Conjeaud, M., S. Harar, M. Mostefai, E.C. Pollaco, C. Volant, Y. Cassagnou, R. Dayras, R. Legrain, H. Oeschler, and F. Saint-Laurent (1985). Phys. Lett., 159B, 244. Csernai, L. P. (1985). Phys. Rev. Lett., 54, 630. Csernal, L. P., and J. Kapusta (1986). Phys. Reports, 131, 223. Cugnon, J., T. Mizutanl, and J. Vandermeulen (1981). Nucl. Phys., A352, 505. Cugnon, J., and M. Jaminon (1983). Phys. Lett., 123B, 155. Cugnon, J. (1984). Lectures given at the Carg~se Summer School. Curtin, M. W., H. Tokl, and D. K. Scott (1983). Phys. Lett., 123B, 289. Danielewicz, P. (1979). Nucl. Phys., A314, 465. Das Gupta, S., and A. Z. Mekjian (1981). Phys. Reports, 72, 131. Devl, K. R. S., M. R. Strayer, K. T. R. Davies, S. E. Ko---onin, and A. K. Dhar (198i). Phys. Rev., C24, 2521. Duek, E., L. Kowalski, M. Rajagopalan, J. M. Alexaner, D. Logan, M. S. Zisman, and M. Kaplan (1982). Z. Phys., A307, 221. FAI, G., and J. Randrup'-~82). Nucl. Phys., A381, 557. Fan, G.-Y., H. Ho, P. L. Gonthler, W. KUhn, A. Pfoh, L. Schad, R. Wolski, J. P. Wurm, J. C. Adloff, D. Dlsdier, V. Rauch, and F. Scheibling (1983). Z. Phys., A310, 269. Fatyga, M., K. Kwlatkowskl, V. E. Viola, C. B. Chltwood, D. J. Fields, C. K. Gelbke, W. G. Lynch, J. Pochodzalla, M. B. Tsang, and M. Blann (1985). Phys. Rev. Lett., 55, 1376. Fields, D. J., W. G. Lynch, C. B. Chltwood, C. K. Gelbke, M. B. Tsang, H. Utsunomlya, and J. Aichelln (1984). Phys. Rev., C30, 1912. Fields, D. J., W. G. Lynch, T. K. Nayak, M. B. Tsang, C. B. Chltwood, C. K. Gelbke, R. Morse, J. Wilczynski, T. C. Awes, R. L. Ferguson, F. Plasll, F. E. Obenshain, and G. R. Young (1986a). Phys. Rev., C34, 536. Fields, D. J., C. K. Gelbke, W. G. Lynch, and J. Pochodzalla (1986b). Michigan State University preprint. Finn, J. E., S. Agarwal, A. BuJak, J. Chuang, L. J. Gutay, A. S. Hirsch, R. W. Minich, N. T. Porile, R. P. Scharenberg, B. C. Stringfellow, and F. Turkot (1982). Phys. Rev. Lett., 49, 1321. Fisher, M.E. (1967). Physics, 3, 255. Fox, D., D. A. Cebra, Z. M. Koenig, P. Ugorowski, and G. D. Westfall (1986). Phys. Rev., C33, 1540. Frankel, K. A., and J. D. Stevenson (1981). Phys. Rev., C23, 1511. Friedman, W. A., and W. G. Lynch (1983a). Phys. Rev., C28, 16. Friedman, W. A., and W. G. Lynch (1983b). Phys. Rev., C ~ , 950. Friedman, W. A. (1984). Phys. Rev., C29, 139. Fukuda, T., M. Ishlhara, H. Ogata. M. Miura, T. Shlmoda, K. Katorl, S. Shimoura, M.-K. Tanaka, E. Takada, and T. Otsuka (1984a). Nucl. Phys., A425, 548. Fukuda, T., T. Nomura, T. Shimoda, K. Katori, S. Shlmoura, K. Suekl, and H. Ogata (1984b). Nucl. Phys., A429, 193. Fulmer, C. B., J. B. Ball, R. L. Ferguson, R. L. Robinson, and J. R. Wu (1981). Phys. Lett., 100B, 305. Fung, S. Y., W. Gorn, G. P. Kiernan, J. J. Lu, Y. T. Oh, and T. T. Poe (1978). Phys. Rev. Lett., 41, 1592. Gaidos, J. A., L. J. Gutay, A. S. Hirsch, R. Mitchell, T. V. Ragland, R. P. Scharenberg, F. Turkot, R. B. Willmann, and C. L. Wilson (1979). Phys. Rev. Lett., 42, 82. Gale, C., and S. Das Gupta (1984). Phys. Rev., C29, 1339. Gale, C., and S. Das Gupta (1985). Phys. Lett., I-~-~2B, 35. Galin, J., H. Oeschler, S. Song, B. Borderle, M. F. Rivet, I. Forest, R. Bimbot, D. Gard~s, B. Gatty, H. Gulllemont, M. Lefort, B. Tamain, and X. Tarrago (1982). Phys. Rev. Lett., 48, 1787. Galin, J. (1985). Nucl. Phys., A447, 519c. Galin, J., G. Ingold, U. Jahnke, D. Hilscher, M. Lehmann, H. Rossner, E. Schwinn and P. Zank (1985). Annual Report, Hahn-Meltner-Institut fur Kernforschung, Berlin, pp. 67-68. Gardner, E., and C. M. G. Lattes (1948a). Phys. Rev., __74, 235A. Gardner, E., and C. M. G. Lattes (1948b). Phys. Rev., 74, 1558A. Garpman, S. I. A., D. Sperber, and M. Zielinska-Pfabe ~980a). Phys. Lett., ~90B, 53. Garpman, S. I. A., S. K. Samaddar, D. Sperber, and M. Zielinska-Pfabe (1980b). Phys. Lett., 92B, 56. Gavron, A., R. L. Ferguson, F. E. Obenshaln, F. P l a s i l , G. R. Young, G. A. P e t i t t , K. Geoffroy Young, D. O. Sarantites, and C. F. Magulre (1981a). Phys. Rev. L e t t . , 46, 8. Gavron, A., J. R. Beene., R. L. Ferguson, F. E. Obenshaln, F. P l a s i l , G. R. Young, G. A. P e t i t t , K. Geoffroy-Young, M. J~skel~inen, D. G. Sarantites, and C. F. Maguire (1981b). Phys. Rev., C24, 2048. Gavron, A., J. R. Beene, B. Cheynls, R. L. Ferguson, F. E. Obenshaln, F. P l a s i l , G. R. Young, G. A. P e t l t t , C. F. Magulre, D. G. S a r a n t i t e s , M. J~skel~inen, and K. Geoffroy-Young (1983). Phys. Rev., C27, 450.

Nucleus - Nucleus Collisions

101

Geoffroy, K. A., D. G. Sarantites, M. L. Halbert, D. C. Hensley, R. A. Dayras, and J. H. Barker (1979). Phys. Hey. Lett., 43, 1303. Geoffroy-Young, K., D. G. Sarantites, J. R. Beene, M. L. Halbert, D. C. Hensley, R. A. Dayras, and J. H. Barker (1981). Phys. Rev., C23, 2479. Gersohel, C., A. Gilbert, N. Perrin, and H. Tricolre (1985). Z. Phys., A322, 433. Glasow, R., G. Gaul, B. Ludewigt, R. Santo, H. Ho, W. KGhn, U. Lynen, and W. F. W. MUller (1983). Phys. Lett., I02B, 71. Gonthler, P. L., H. Ho, M. N. Namboodiri, J. B. Natowitz, L. Adler, S. Simon, K. Hagel, S. Kniffen, and A. Khodai (1983). Nucl. Phys., A411, 289. Goldhaber, G., S. Goldhaber, W. Lee, and A. Pais (1960). Phys. Rev., 120, 300. Goldhaber, A. S. (1978). Phys. Rev., __C17, 2243. Goodman, A. L., J. I. Kapusta, and A. Mekjian (1984). Phys. Rev., C30, 851. Gosset, J., H. H. Gutbrod, W. G. Meyer, A. M. Poskanzer, A. Sandoval, R. Stock, and G. D. Westfall (1977). Phys. Rev., C16, 1977. Gosset, J., J. I. Kapusta, and G.D. Westfall (1978). Phys. Rev., C18, 844. Gottschalk, P. A., and M. WestrOm (1977). Phys. Rev. Lett., 39, 1250. Gottschalk, P. A., and M. WestrOm (1979). Nucl. Phys., A314, 232. Green, R. E. L., and R. G. Kortellng (1980). Phys. Rev., C22, 1594. Gr@golre, C., and F. Scheuter (1984). Phys. Lett., 146B, Gr@golre, C., B. Remaud, F. Scheuter, and F. S@bille (1985). Nucl. Phys., A436, 365. Gr@goire, C., B. Remaud, F. S6bille, L. Vinet, and Y. Raffray (1986). GANIL preprint. Grosse, E. (1985). Nucl. Phys., A447, 611c. Grosse, E., P. Grimm, H. Heckwolf, W. F. J. MOller, H. Noll, A. Oskarson, H. Stelzer, and W. ROsch (1986). Europhys. Lett., ~, 9. Gross, D. H. E., L. Satpathy, Meng Ta-chung, and M. Satpathy (1982). Z. Phys., A309, 41. Gross, D. H. E., and Xiao-ze Zhang (1985). Phys. Lett., 161B, 47. Gross, D. H. E., Zhang Xiao-ze, and Xu Shu-yan (1986). Phys. Rev. Lett., 56, 1544. Guet, C. and M. Prakash (1984). Nucl. Phys., A428, 119c. Gustafsson, H. A, H. H. Gutbrod, B. Kolb, H. LOhner, B. Ludewlgt, A. M. Poskanzer, T. Renner, H. Riedesel, H. G. Ritter, A. Warwick, F. Welk, and H. Wieman (1984). Phys. Roy. Lett., 53, 544. Gutbrod, H. H., A. Sandoval, P. J. Johansen, A. M. Poskanzer, J. Gosset, W. G. Meyer, G. D. Westfall, and R. Stock (1976). Phys. Roy. Lett., 37. 667. Gutbrod, H. H., A. I. Warwick, and H. Wieman (1982). Nucl. Phys., A387, 177c. Hahn, D., and H. StOcker (1986). Lawrence Berkeley Laboratory Report, LBL-21386. Hale, G. M., and B. C. Dodder (1984). Few-Body Problems in Physics, edited by B. Zeltnltz (Elsevier, Amsterdam, 1984) Vol. 2, p. 433. Hanbury-Brown, R., and R.Q. Twlss (19~). Nature, 178, 1046. Hasselqulst, B. E., G. M. Crawley, B. V. Jacak, Z. M. Koenlg, G. D. Westfall, J. E. Yurkon, R. S. Tickle, J. P. Dufour, and T. J. M. Symons (1985). Phys. Rev., C32, 145. Hauser, W., and H. Feshbach (1952). Phys. Rev., 87, 366. Heckwolf, H., E. Grosse, H. Dabrowskl, O. Klepper, C. Michel, W. F. J. MOller, H. No11, C. Brendel, W. RGsch, J. Jullen, G. S. Pappalardo, G. Bizard, J. L. Lavllle, A. C. Mueller, and J. P~ter (1984). Z. Phys., A315, 243. Hirsch, A. S., A. BuJak, J. E. Finn, L. J. Gutay, R. W. Minlch, N. T. Porile, R. P. Scharenberg, and B. C. Strlngfellow (1984). Phys. Rev., C29, 508. Ho, H., P. Gonthler, M. N. Namboodlrl, J. B. Natowltz, L. Adler, S. Simon, K. Hagel, R. Terry, and A. Khodal (1980). Phys. Lett., 93_._~B,51. Holub, E., D. Hilscher, G. Ingold, U. Jahnke, H. Off, and H. Rosner (1983a). Phys. Rev., C28, 252. Holub, E., M. Korollja, and N. Cindro (1983b). Z. Phys., A314, 347. Holub, E., D. Hilscher, G. Ingold, U. Jahnke, H. Off, H. Rosner, W. P. Zank, W. U. ScbrOder, H. Gemmeke, K. Keller, L. Lassen, and W. L~cklng (1986). Phys. Rev., C3__~3, 143. HUfner, J. and D. Mukhopadhyay (1986). Phys. Lett., 173B, 373. Hulzenga, J., J. R. Birkelund, L. E. Tubbs, D. Hilscher, U. Jahnke, H. Rossner, B, Gebauer, and H. Lettau (1983). Phys. Rev., C28, 1853. Ikezoe, N. Shlkazono, Y. Tomlta, K. Ideno, Y. Suglyama, and E. Takekoshl (1985). Nucl. Phys., A444, 349. Inamura, T., M. Ishlhara, T. Fukuda, T. Shlmoda, and H. Hiruta (1977). Phys. Lett., 68B, 51. Inamura, T., T. Kojlma, T. Nomura, T. Sugltate, and B. Utsunomlya (1979). Phys. Left., 84B, 71. Jacak, B. V., G. D. Westfall, C.K. Gelbke, L. H. Harwood, W. G. Lynch, D. K. Scott, H. StOcker, M. B. Tsang, and T. J. M. Symons (1983). Phys. Roy. Lett., 51, 1846. Jacak, B. V., H. StOcker, and G. D. Westfall (1984). Phys. Rev., C29, 1744. Jacak, B. V., D. Fox, and G. D. Westfall (1985). Phys. Rev., C31,-~4. Jacquet, D., E. Duek, J. M. Alexander, B. Borderle, J. Galln, D. Oard~s, D. Guerreau, M. Lefort, F. Monnet, M. F. Rivet, and X. Tarrago (1984). Phys. Rev. Lett., 53, 2226.

102

Claus-Konrad Gelbk¢ and David H. Boal

J a e q u e t , D., J . O a l i n , M. F. R i v e t , R. Bimbot, B. Borderie, D. Gard~s, B. Garry, D. Guerreau, L. Kowalskl, M. Lefort, and X. Tarrago (1985a). Nucl. Phys., A455, 140. Jacquet, D., J. Galin, B. Borderle, D. Gard~s, D. Guerreau, M. Lefort, M. Monnet, M. F. Rivet, X. Tarrago, E. Duek, and J. M. Alexander (1985b). Phys. Bey., C322, 1594. Jakobsson, B., L. Carl~n, P. Kristiansson, J. Krumlinde, A. Oskarsson, I. Otterlund, B. Schr~der, H.-R. Gustafsson, T. Johansson, H. Ryde, G. Tibell, J. P. Bondorf, G. F~I, A. O. T. Karvinen, O. B. Nielsen, M. Buenerd, J. Cole, D. Lebrun, J. M. Loiseau, P. Martin, R. Ost, P. de Salntignon, C. Guet, E. Monnand, J. Mougey, H. Nifenecker, P. Perrin, J. Pinston, C. Ristorl, and F. Schussler (1981). Phys. Left., I02B, 121. Jaqaman, H. R., A. Z. MekJian, and L. Zamick (]983). Phys. Rev., C27, 2782. Jaqaman, H. R., A. Z. MekJian, and L. Zamick (1984). Phys. Rev., C29, 2067. Jennlngs, B. K., S. Das Gupta, and N. Mobed (1982). Phys. Rev., C25, 278. Jennings, B. K., D. H. Boal, and J. C. Shillcock (1986). Phys. Rev., C33, ]303. Johansson, T., H.-R. Gustafsson, B. Jakobsson, P. Kristlansson, B. Nor~n, A. Oskarsson, L. Carl~n, I. Otterlund, H. Ryde, J. Julien, C. Guet, R. Bertholet, M. Maurel, H. Nifeneeker, P. Perrin, F. Bchussler, G. Tibell, M. Buenerd, J. M. Lioseaux, P. Martin, J. P. Bondorf, O.-B. Nielsen, A. O. T. Karvinen, and J. MouEey (]982). Phys. Rev. Lett., 48, 732. Kadanoff, L. P., and G. Baym (1962). Quantum Statistical Mechanics, Benjamin, New York. Kapusta, J. I. (1980). Phys. Rev., C2_!, 1301. Kapusta, J. (1984). Phys. Rev., C29, 1735. Karvinen, A. O. T., J. N. De, and B. Jakobsson (1981). Nucl. Phys., A367, 122. Kasagi, J., S. Salni, T. C. Awes, A. Galonsky, C. K. Gelbke, G. Poggi, D. K. Scott, K. L. Wolf, and R. L. Legrain (1981). Phys. Left., IO~B, 434. Kikuchi, K., and M. Kawai (1968). Nuclear Reactions at High Energy, North Holland, Amsterdam. Knoll, J. (1979), Phys. Rev., C20, 773. Knoll, J. (1980). Nucl. Phys., A343, 511. Knoll, J., and B. Strack (1984). Phys. Lett., 149B, 45. Ko, C. M., G. Bertsch, and J. Aichelin (1985). Phys. Rev., C31, 2324. K6hler, H. S. (1982). Nucl. Phys., A378, 159. KOnler, H. S., and B. S. Nilsson (19-~'~. Nuel. Phys., A417, 541. Koonin, S. E. (1977). Phys. Lett., __70B, 43. Kopylov, G. I. (1974). Phys. Lett., 50B, 472. Kopylov, G. I., and M. i. Podgoretskii (1974). Soviet J. Nucl., 18, 336. Krebs, G. F., J.-F. Gilot, P. N. Kirk, G. Claesson, J. Miller, H. G. Pugh, L. S. Schroeder, G. Roche, J. Vicente, K. Beard, W. Benenson, J. van der Plicht, B. Sherrill, and J. W. Harris (1986). Phys. Left., 171B, 37. Kristiansson, P., L. Carl-'~-~, H.-R. Gustafsson, B. Jakobsson, A. Oskarsson, H.Ryde, J. P. Bondorf, O.-B. Nielsen, G. L6vh6iden, T.-F. Thorsteinsen, D. Heuer, and H. Nifencker (1985). Phys. Lett., 155B, 31. Kruse, H., B. V. Jacak, and H. StOcker (1985a). Phys. Rev. Left., __54, 289. Kruse, H., B. V. Jacak, J. J. Molitoris, G. D. Westfall, and H. StOcker (1985b). Phys. Rev., 3__.!, C 1770. Kwato NJock, M., M, Maurel, E. Monnand, H. Nifenecker, J. Pinston, F. Sehussler, and D. Barneoud (1986). Phys. Left., 175B, 125. Kwiatkowski, K., J. Bashkin, H. Karwowski, M. Fatyga, and V. E. Viola (1986). Phys. Left., 171B, 41. Kyanowski, A., F. Saint-Laurent, D. Ardouin, H. Delagrange, H. Doubre, C. Gr~goire, W. Mittig, A. P~ghaire, J. P~ter, Y. P. Viyogi, B. Zwieglinkei, J. Qu~bert, G. Bizard, F. Lef~bvres, B. Tamain, J. Pochodzalla, C. K. Gelbke, W. Lynch, and M. Maier (1986). Phys. Left., 181B, 43. Lamb, D. Q., J. M. Lattimer, C. J. Pethick, and D. G. Ravenhall (1978). Phys. Rev. Lett., 41, 1623. Landau, L. D. (1936). Phys. Z. Soy 4. Union, 10, 154. La Rana, G., G. Nebbia, E. Tomasl, C. NgS, X. S. Chen, G. Nebbia, P. Lhenoret, R. Lueas, C. Mazur, M. Ribrag, C. Cerrutl, S. Chiodelli, A. Demeyer, D. Guinet, J. L. Charvet, M. Morjean, A. P~ghaire, Y. Pranal, L. Sinopoli, J. Uzureau, and R. de Swinlarskl (1983). Nucl. Phys., A407, 233. Laville, J. L., C. Le Brun, J. F. Leoolley, F. Lef~bvres, M. Louvel, R. Regimbart, J.C. Steokmeyer, N. Jabbri, R. Bertholet, C. Guet, D. Heuer, M. Maurel, H. Nifenecker, C. Ristori, F. Schussler, F. Guilbault, and C. Lebrun (1984). Phys. Left., 138B, 35. Lemaire, M.-C., S. Nagamiya, S. Schnetzer, H. Steiner, and I. Tanihata (197-~9--~.Phys. Left.,

85....~B, 38. L e r a y , S . , O. G r a n i e r , C. NgS, E. T o m a s i , C. C e r r u t i , P. L ' H e n o r e t , R. Lucas, C. Mazur, M. R i b r a g , J.L. Charvet, C. Hunneau, J.P. Lochard, M. MorJean, Y. Patln, L. Signopoli, J. Uzureau, D. Guinet, L. Vagneron, and A. P6ghaire, (1985). Z. Phys., A320, 533. Levit, S., and P. Bonche (1985). Nucl. Phys., A~37, 426. Lopez, J. A., and P. J. Siemens (1984). Nucl. Phys., A431, 728.

Nucleus - Nucleus Collisions

103

hu, d. d., D. Beavls, S. Y. Fung, W. G e m , A. Hule, G. P. Kiernan, R. T. Poe, and D. VanDahlen (1981). Phys. Rev. h e t t . , 46, 898. Lynch, W. O., L. W. Richardson, M. B. Tsang, R. E. E l l i s , C. K. Gelbke, and R. E. Warner (1982). Phys. L e t t . , I08B, 274. Lynch, W. O., C. B. Chltwood, M. B. Tsang, D. J. F i e l d s , D. R. Klesoh, C. K. Gelbke, O. R. Young, T. C. Awes, R. L. Ferguson, F. E. Obenshaln, F. P l a s i l , R. L. Robinson, and A. D. Panaglotou (1983). Phys. Rev. L e t t . , __51, 1850. Lynen, U., H. Ho, W. KOhn, D. Pelte, U. Winkler, W. F. J. MUller, Y.-T. Chu, P. Doll, A. Gobbi, K. Hildenbrand, A. Olml, H. Sann, H. Stelzer, R. Book, H. Lohner, R. Glsow, and R. Santo (1982). Nuel. Phys., A387, 129o. McMahan, M. A., L. G. Moretto, M. L. Padgett, G. J. Wozniak, L. G. Sobotka, and M. G. Mustafa (1985). Phys. Rev. Lett., 54, 1995. MekJian, A. (1977). Phys. Bey. Lett., 38, 640. Mekjian, A. (1980). Phys. Lett., 89B, 177. Mlnich, R. W. S. Agarwal, A. BuJak, J. Chuang, d. E. Finn, L. d. Outay, A. S. Hirsoh, N. T. Porile, R. P. Soharenberg, B. C. Stringfellow, and F. Furkot (1982). Phys. Lett., 118B, 458. Mittig, W., A. Cunsolo, A. Fotl, d. P. Wleleczko, F. Auger, B. Berthier, d. M. Pasoaud, d. Qu~bert, and E. Plagnol (1985). Phys. Lett., 154B, 259. Molitoris, d. d., D. Hahn, and H. StOoker (1985). Nucl. Phys., A447, 130. Molitoris, d. d., and H. StOoker (1985). Phys. Lett., 162B, 47. Mooney, P., W. W. Morlson, S. K. Samaddar, D. Sperber, and M. Zielinska-Pfabe (1981). Phys. Lett., 98B, 240. Moretto, L. G. (1975). Nuel. Phys., A247, 211. Morison, W. W., S. K. Samaddar, D. Sperber, and M. Zielinska-Pfabe (1980). Phys. Lett., 93B, 379. Morgenstern, H., W. Bohne, K. Orabisch, D. O. Kovar, and H. Lehr (1982). Phys. Lett., 133B, 463. Morgenstern, H., W. Bohne, K. Grabisoh, H. Lehr, and W. StOffler (1983). Z. Phys., A313, 39. Morgenstern, H., W. Bohne, W. Galster, K. Grablsch, and A. Kyanowskl (1984). Phys. Bey. Lett.,

5._.22, 11o4. Morrlssey, D. J., W. Benenson, E. Kashy, B. Sherrlll, A. D. Panaglotou, R. A. Blue, R. M. Ronnlngen, J. van de Plicht, and H. Utsunomiya (1984). Phys. Lett., 148B, 423. Morrlssey, D. J., W. Benenson, E. Kashy, C. Bloch, M. Lowe, R. A. Blue, R. M. Ronningen, B. Sherrlll, H. Utsunomiya, and I. Kelson (1985). Phys. Roy., C32, 877. Morrissey, D. J., C. Bloch, W. Benenson, E. Kashy, R. A. Blue, R. M. Ronningen, and R. AryaeineJad (1986). Phys. Bey., C34, 761. Mosel, U. (1984). In A. D. Bromley (Ed.), Treatease on Heavy Ion Science, Vol. 2, Plenum, New York, p. 3. Nagamiya, S., M.-C. Lemalre, E. Moeller, S. Schnetzer, O. Shapiro, H. Stelner, and I. Tanlhata (1981). Phys. Rev., C2.__~4,971. Nagamlya, S. (1982). Phys. Rev. Lett., 49, 1383. Nagamlya, S., H. Hamagaki, P. Heoklng, S. Kadota, R. Lombard, Y. Miake, E. M o e l l e r , S. Schnetzer, H. Stelner, I. Tanlhata, S. Bohrmann, and J. Knoll (1982). Phys. Rev. Lett.,

48B, 1780. Nagamlya, S. and M. Gyulassy (1984). Advances in Nuclear Physics, Vol. 13, Plenum, New York, p. 201. NgS, C., and S. Leray (1985). Z. Phys., A322, 419. Nifenecker, H., and J. P. Bondorf (1985). Nucl. Phys., A442, 478. Nix, J. R., (1979). In D. Wilkinson (Ed.), Pro~. Part. Nucl. Phys., Vol. 2, Pergamon, Oxford, p. 237. Nell, H., E. Grosse, P. Braun-Munzinger, H. Dabrowski, H. Heckwolf, O. Klepper, C. Michel, W. F. d. MUller, H, Stelzer, C. Brendel, and W. BOsch (1984). Phys. Bey. Lett., 52, 1284. Nomura, T., H. Utsunomlya, T. Motobayashi, T. Inamura, and M. Yanokura (1978T?. Phys. Bey. Lett., 40, 694. Nordhelm, L. W. (1928). Prec. Roy. See., A119, 689. Panaglotou, A. D., M. W. Curtin, H. Toki, D. K. Scott, and P. J. Siemens (1984). Phys. Bey. Lett., 52, 496. Panagiotou, A. D., M. W. Curtin, and D. K. Scott (1985). Phys. Rev., C31, 55. Pochodzalla, J., W. A. F r i e d m a n , C. K. Oelb~e, W. O. Lynch, M. Major, D. Ardouin, H. Delagrange, H. Doubre, C. Gr~goire, A. Kyanowski, W. Mittig, A. P~ghaire, J. P~ter, F. Saint-Laurent, Y. P. Viyogl, B. Zwieglinskl, G. Bizard, F. Lef~bvres, B. Tamaln, and J. Qu~bert (1985a). Phys. Rev. Lett., 55, 177. Pochodzalla, J., W. A. F r i e d m a n , C. K. Oelbke, W. 0. Lynch, M. Maier, D. Ardouln, H. Delagrange, H. Doubre, C. Gr~goire, A. Kyanowski, W. Mittig, A. P~ghaire, d. P~ter, F. Saint-Laurent, Y. P. Viyogi, B. Zwieglinski, G. Bizard, F. Lef~bvres, B. Tamain, and J. Qu~bert (1985b). Phys. Lett., 161B, 256.

104

C l a u s - K o n r a d G e l b k e a n d D a v i d H . Boal

Poohodzalla, J., W. A. F r i e d m a n , C. K. Gelbke, W. G. Lynch, M. Haler, D. Ardouin, H. Delagrange, H. Doubre, C. Gr@golre, A. Kyanowski, W. Mittig, A. P~ghalre, J. P@ter, F. Saint-Laurent, Y. P. ViyogI, B. Zwieglinski, G. Bizard, F. Lef~bvres, B. Tamain, and J. Qu~bert (19850). Phys. Lett., 161B, 275. Pochodzalla, J., C. B. Chitwood, D. J. Fields, C. K. Gelbke, W. G. Lynch, M. B. Tsang, D. H. Boal, and J. C. Shillcook (1986a). Phys. Lett., 174B, 36. Poohodzalla, J., C. K. Gelbke, C. B. Chitwood, D. J. Fields, W. G. Lynch, M. B. Tsang, and W. A. Friedman (1986b). Phys. Left., 175B, 275. Poohodzalla, J., C. K. Gelbke, W. G. Lynch, M. Maier, D. Ardouin, H. Delagrange, H. Doubre, C. Gr~goire, A. Kyanowski, W. Mittig, A. P ~ g h a i r e , J. P~ter, F. S a i n t - L a u r e n t , B. Zwiegllnski, G. Bizard, F. Lef~bvres, B. Tamain, and J. Qu~bert, Y. P. Viyogi, W. A. Friedman, and D. H. Boal (1986e). Michigan State University preprint. Pollaco, E. C., M. Conjeaud, S. Harar, C. Volant, Y. Casagnou, R. Dayras, R. Legrain, M. S. Nguyen, H. Oeschler, and F. Saint-Laurent (1984). Phys. Lett., 146B, 29. Poskanzer, A. M., G. W. Butler, and E. K. Hyde (1971). Phys. Roy., C_~3, 882. Prakash, M., P. Braun-Munzinger, and J. Staehel (1986). Phys. Rev., C33, 937. Pratt, S. (1984). Phys. Rev. Lett., 53, 1219. Randrup, J., and S. E. Koonin (1981). Nuol. Phys., A356, 223. Reinhardt, R., and H. Sohulz (1985). Nucl. Phys., A432, 630. Remaud, B., F. S@bille, C. Gr~goire, L. Vinet, and Y. Raffray (1985). Nucl. Phys., A432, 630. Remington, B. A., G. Caskey, A. Galonsky, C. K. Gelbke, L. Heilbronn, J. Heltsley, M. B. Tsang, F. Deak, A. Kiss, Z. Seres, J. Kasagi, and J. J. Kolata (1986). Phys. Rev., C34, 1685. Remler, E. A. (1981). Ann. Phys. (N.Y.), 136, 293. Robinson, R. L., R. L. Auble, I. Y. Lee, M. J. Martin, G. R. Young, J. Gomez del Campo, J. B. Ball, F. E. Bertrand, R. L. Ferguson, C. B. Fulmer, J. R. Wu, J. C. Wells, and H. Yamada ( 1 9 8 1 ) . P h y s . R e v . , C2~4, 2084. R o b i n s o n , R. L . , R. L. A u b l e , J . B. B a l l , F. E. B e r t r a n d , R. L. F e r g u s o n , C. B. F u l m e r , J . Gomez del Campo, I. Y. Lee, R. W. Novotny, G. R. Young, J. C. Wells, C. F. Maguire, and H. Yamada (1983). Phys. Roy., __C27, 3006. R~pke, G. (1983). Phys. Lett., 121B, 223. R~pke, G., H. Schulz, L. N. Andronenko, A. A. Kotov, W. Neubert, and E. N. Volnin (1985). Phys. Rev., C31, 1556. Rosner, G., J. Poehodzalla, H. Beck, G. Hlawatsch, A. Miczaika, H. J. Rabe, R. Butsch, B. Kolb, and B. Sedelmeyer (1985). Phys. Left., 150B, 87. Sagawa, H., and G. F. Bertseh (1985). Phys. Lett., 155B, 11. Sate, H., and K. Yazaki (1981). Phys. Lett., __98B, 153. Sate, H., and K. Yazaki (1980). In K. Nakai and A. S. Goldhaber (Ed.), "Hi~h-Ener~y Nuclear Interactions and Properties of Dense Nuclear Matter", ProceedinGs of the Hakone Seminar, p. 340. Sauer, G., H. Chandra, and U. Mosel (1976). Nucl. Phys., A264, 221. Schr~der, W. U., and J. R. Hulzenga (1984). In A. D. Bromley (Ed.), Treatease on Heavy Ion Science, Vol. 2, Plenum, New York, p. 115. Schulz, H., L. MOnehow, G. ROpke, and M. Schmidt (1982). Phys. Left., I19B, 12. Schwarzschild, A., and C. Zupanclc (1963). Phys. Roy., 129, 854. S~bllle, F, and B. Remaud (1984). Nucl. Phys., A420, 141. Shuryak, E. V. (1973). Phys. Left., 44B, 387. Shyam, R., and J. Knoll (1984a). Nucl. Phys., A246, 606. Shyam, R., and J. Knoll (1984b). P~ys. Left., 136B, 221. Shyam, R., and J. Knoll (1986). Erratum. Siemens, P. J., and J. Kapusta (1979). Phys. key. Lett., 43, 1486. Siemens, P. J., and J. O. Rasmussen (1979). Phys. Rev. Lett., 42, 880. Siemens, P. J. (1983). Nature, 305, 410. Slemssen, R. H., G. J. Balster, H. W. Wilschut, P. D. Bond, P. C. N. Crouzen, P. B. Goldhoorn, Han Shukul, and Z. Suikowski (1985). Phys. Lett., 161B, 261. Sikkeland, T., E. L. Haines, and V. E. Viola, Jr. (1962), Phys. Roy., 125, 1350. Siwek-Wilczy~ska, K., E. H. du Marchle van Voorthuysen, J. Van Popta, R. H. Siemssen, and J. Wilczy~ski (1979a). Phys. Roy. Lett., __42, 1599. Siwek-Wilczy~ska, K., E. H. du Marchie van Voorthuysen, J. Van Popta, R. H. Siemssen, and J. Wilczy~ski (1979b). Nucl. Phys., A330, 150. Sobel, M. I., P. J. Siemens, J. P. Bondorf, and H. A. Bethe (1975). Nuel. Phys., A251, 502. Sobotka, L. G., M. L. Padget, G. J. Wozniak, G. Guarino, A. J. Paeheco, L. G. Moretto, Y. Chan, R. G. Stokstad, I. Tserruya, and S. Wald (1983). Phys. Roy. Left., 5~I, 2187. Sobotka, L. G., M. A. McMahan, R. J. McDonald, C. Signarbieux, G. J. Wozniak, M. L. Padgett, J. H. Gu, Z. H. Liu, Z. Q. Tao, and L. G. Moretto (1984). Phys. Rev. Left., __53, 2004. Song, S., M. F. Rivet, R. Bimbot, B. Borderie, I. Forest, J. Galin, D. Gard~s, B. Gatty, M. Lefort, H. Oeschler, B. Tamaln, and X. Tarrago (1983). Phys. Lett., 130B, 14.

Nucleus - Nucleus Collisions

105

Staehel, J., P. Braun-Munzlnger, R. H. Frelesleben, P. Paul, S. Sen, P. De Young, P. H. Zhang, T. C. Awes, F. E. Obenshain, F. Plasil, G. R. Young, R. Fox, and R. Ronnlngen (1986). Phys. Rev., C33, 1420. Stelte, N., M. WestrOm, and R. M. Weiner (1982). Nucl. Phys., A384, 190. Stephans, G. S. F., D. G. Kovar, R. V. F. Janssens, G. Rosner, H. Ikezoe, B. Wilklns, D. Henderson, K. T. Lesko, J. J. Kolata, C. K. Gelbke, B. V. Jaeak, Z. M. Koening, G. D. Westfall, A. Szanto de Toledo, E. M. Szanto, and P. L. Gonthier (1985). Phys. Lett., 161B, 60. Stevenson, g., K. B. Beard, W. Benenson, J. Clayton, E. Kashy, A. Lampis, D. J. Morrissey, M. Samuel, R. J. Smith, C. L. Tam, and J. S. Winfield (1986). Phys. Rev. Lett., 57, 555. Stock, R. (1986). Phys. Reports, 135, 259. StGeker, H., A. A. Ogloblln, and W. Grelner (1981). Z. Phys., A313, 259. StGcker, H., G. Buchwald, G. Graebner, P. Subramanlan, J. A. Maruhn, W. Grelner, B. V. Jaoak, and G. D. Westfall (1983). Nucl. Phys., A400, 63c. StGcker, H., and W. Greiner (1986). Phys. Reports, 137, 277. Strack, B., and J. Knoll (1984). Z. Phys., A315, 249. Symons, T. J. M., P. Doll, M. Binl, D. L. Hendrle, J. Mahoney, G. Mantzouranis, D. K. Scott, K. van Bibber, Y. P. Viyogl, H. H. Wleman, and C. K. Gelbke (1980). Phys. Lett., 94B, 131. Tanlhata, I., M.-C. Lemaire, S. Nagamiya, and S. Sehnetzer (1980). Phys. Lett., 97B, 363. Tanlhata, I., S. Nagamlya, S. Schnetzer, and H. Steiner (1981). Phys. Lett., IOOB, 121. Trautmann, W., O. Hansen, H. Trlcolre, W. Hering, R. Ritzka, and W. Tromblk [T984). Phys. Rev. Lett., 53, 1630. Tricolre, H., C. Gerschel, A. Gilbert, and N. Perrin (1986). Z. Phys., A323, 163. Trockel, R., K. D. Hildenbrand, U. Lynen, W. F. J. Mueller, H. J. Rabe, H. Sann, H. Stelzer, R. Wada, N. Brummund, R. Glasow, K. H. Kampert, R. Santo, D. Pelte, J. Pochodzalla, and E. Eckert (1985). Report GSI-85-45. Tsang, M. B., D. R. Klesch, C. B. Chitwood, D. J. Fields, C. K. Gelbke, W. G. Lynch, H. Utsunomiya, K. Kwiatkowski, V.E. Viola, Jr., and M. Fatyga (1984a). Phys. Lett., 134B, 169. Tsang, M. B., C. B. Chitwood, D. J. Fields, C. K. Gelbke, D. R. Klesch, W. G. Lynch, K. Kwiatkowskl, and V. E. Viola, Jr., (1984b). Phys. Rev. Lett., 52, 1967. Tsang, M. B., W. G. Lynch, C. B. Chitwood, D. J. Fields, D. R. Klesch, C. K. Gelbke, G. R. Young, T. C. Awes, R. L. Ferguson, F. E. Obenshain, F. Plasil, and R. L. Robinson (19840). Phys. Lett., 148B, 265. Tsang, M. B., R. M. Ronningen, G. Bertseh, Z. Chen, C. B. Chltwood, D. J. Fields, C. K. Gelbke, W. G. Lynch, T. Nayak, J. Pochodzalla, T. Shea, and W. Trautmann (1986). Phys. Rev. Lett., 57, 559. Tubbs, L. E., J. R. Birkelund, J. R. Hulzenga, D. Hilseher, U. Jahnke, H. Rossner, and B. Gebauer (1985). Phys. Rev., C32, 214. Uehling, E. A., and G. E. Uhlenbeck (1933). Phys. Rev. 43, 552. Umar, A. S., M. R. Strayer, and D. J. Ernst (1984). Phys. Lett., I04B, 290. Utsunomlya, H., T. Nomura, T. Inamura, T. Sugltate, and T. Motobayashl (1980). Nuel. Phys., A334, 127 Utsunomlya, T. Nomura, M. Ishlhara, T. Sugitate, K. Ieki, and S. Kohmoto (1981). Phys. Lett., I05B, 135. Vandenbosch, R., and J. R. Hulzenga (1973). Nuclear Fission, Academic Press. Vasak, D., B. MOller, and W. Greiner (1980). Phys. Scripta, 22, 25. Vasak, D., B. MOller, and W. Grelner (1984). Nucl. Phys., A428, 291c. Vineentinl, A., G. Jacucci, and V. R. Pandharlpande (1985). Phys. Rev., C31, 1783. Viola, V. E., B. B. Back, K. L. Wolf, T. C. Awes, C. K. Gelbke, and H. Breuer (1982). Phys. Rev., C26, 178. Vlasov, A. ~'~38). Zh. Eksp. Teor. Flz., 8, 291. Warwick, A. I., A. Baden, H. H. Gutbrod, M. R. Maier, J. P~ter, H. G. Ritter, H. Stelzer, H. H. Wieman, F. Weik, M. Freedman, D. J. Henderson, S. B. Kaufman, E. P. Steinberg, and B. D. Wilkins (1982). Phys. Rev. Lett., 48, 1719. Warwick, A. I., H. H. Wleman, H. H. Gutbrod, M. R. Maier, J. P~ter, H. G. Ritter, H. Stelzer, F. Welk, M. Freedman, D. J. Henderson, S. B. Kaufman, E. P. Steinberg, and B. D. Wilkins (1983). Phys. Rev., C27, 1083. Welner, B., and M. WeststrGm (1977). Nuel. Phys., A286, 282. Westerberg, L., D. G. Sarantites, D. C. Hensley, R. A. Dayras, M. L. Halbert, and J. H. Barker (1978). Phys. Rev., C18, 796. Westfall, G. D., J. Gosset, P. J. Johansen, A. M. Poskanzer, W. G. Meyer, H. H. Gutbrod, A. Sandoval, and R. Stock (1976). Phys. Rev. Lett., 37, 1202. Westfall, G. D., R. G. Sextro, A. M. Poskanzer, A. M. Zebelman, G. W. Butler, and E. K. Hyde (1978). Phys. Rev., C17, 1368.

106

Claus-Konrad Gelbke and David H. Boal

Westfall, O. D., B. V. Jacak, N. Anantaraman, M. W. Curtin, G. M. Crawley, C. K. Gelbke, B. Hasselqulst, W. G. Lynch, D. K. Scott, M. B. Tsang, M. J. Murphy, T. J. M. Symons, R. Legraln, and T. J. Majors (1982). Phys. Lett., 116B, 118. Westfall, G. D., Z. M. Koenlg, B. V. Jacak, h. H. Harwood, G. M. Crawley, M. W. Curtln, C. K. Gelbke, B. Hasselqulst, W. G. Lynch, A. D. Panaglotou, D. K. Scott, H. StScker, and M. B. Tsang (1984). Phys. Rev., C29, 861. Wilczy~skl, J., K. Siwek-Wilezy~s-'ka, J. van Drlel, S. GonggrIjp, D. C. J. M. Hageman, R. V. F. Janssens, J. hukasiak, and R. H. Siemssen (1980). Phys. Rev. Lett., 45, 606. Wilezyfmkl, J., K. Siwek-Wilczy~ska, J. van Driel, S. GonggriJp, D. C. J. M. Hageman, R. V. F. Janssene, J. hukaslak, R. H. Siemssen, and S. Y. van der Werf (1982). Nucl. Phys., A373, 109. Wilets, h., E. M. Henley, M. Kraft, and A. D. MacKellar (1977). Nuol. Phys., A282, 361. Wilklns, B. D., S. B. Kauffman, E. P. Steinberg, J. A. Urbon, and D. J. Henderson (1979). Phys. Rev. Lett., 4.~3, 1080. Xu, H. M., D. J. Fields, W. G. Lynch, M. B. Tsang, C. K. Gelbke, D. Hahn, M. R. Maler, D. J. Morrissey, J. Pochodzalla, H. StOcker, D. G. Sarantites, L. G. Sobotka, M. L. Halbert, and D. C. Hensley (1986). Michigan State University Report, MSUCL-560. Yamada, H., D. R. Zolnowskl, S. E. Cala, A. C. Kahler, J. Pierce, and T. T. Suglhara (1979). Phys. Rev. Lett., 43, 605. Yamada, C. F. Maguire, J. H. Hamilton, A. V. Ramayya, D. C. Hensley, M. L. Halbert, R. L. Robinson, F. E. Bertrand, and R. Woodward (1981).Phys. Rev., C2_._~4,2565. Yano, F. B., and S. E. Koonin (1978). Phys. Lett., 78B, 556. Young, G. R., F. E. Obenshain, F. Plasil, P. Braun-Munzinger, F. Freifelder, P. Paul, and J. Stachel (1986). Phys. Rev., C33, 742. ZaJc, W. A., J. A. Bistlrllch, R. R. Bossingham, H. R. Bowman, C. W. Clawson, K. M. Crowe, K. A. Frankel, J. G. Ingersoll, J. M. Kurck, C. J. Martoff, D. L. Murphy, J. O. Rasmussen, J. P. Sullivan, E. Yoo, O. Hashlmoto, M. Koike, W. J. McDonald, J. P. Miller, and P. Tru~l (1984). Phys. Hey., C29, 2173. Zarbakhsh, F., A. L. Sagle, F. Brochard, T. A. Mulera, V. Perrez-Mendez, R. T a l a g a , I. Tanihata, J. B. Carroll, K. S. Ganezer, G. Igo, J. Ostens, D. Woodard, and I. Sutter (1981). Phys. Rev. Lett., 46, 1268. Zolnowski, D. R., H. Yamada, ~-. E. Cala, A. C. Kahler, and T. T. Sugihara (1978). Phys. Rev. Lett., 41, 92.