Determination of the initial number of degrees of freedom in nucleus-nucleus collisions

Nuclear Physics A487 (1988) 442-456 North-Holland, Amsterdam

DETERMINATION OF THE INITIAL NUMBER OF DEGREES OF FREEDOM IN NUCLEUS-NUCLEUS COLLISIONS M. KOROLIJA, N. CINDRO and R. (2APLAR Ruder Bogkovi6 Institute, 41001 Zagreb, P.O.B. 1016, Croatia, Yugoslavia

R.L. AUBLE, J.B. BALL and R.L. ROBINSON Oak Ridge National Laboratory, Oak Ridge, Tennessee 37831, USA

Received 10 May 1988 Abstract: Inclusive proton spectra from collisions of 32S on Z7AI,46Ti, 6°Ni, 12°Sn, 124Snand 197Auat

503.7 and 678.8 MeV have been analyzed. Contributions from central-like collisions have been singled out using a refined multisource analysis and compared with a Boltzmann master-equation calculation. This comparison yields no, the initial number of degrees of freedom that share the available energy. The dependence of no on various physical quantities has been investigated. A smooth dependence of no- Ap (Ap, the projectile mass number) on the entrance-channel-asymmetry parameter has been observed. I. Introduction

This p a p e r treats equilibrium and preequilibrium nucleon emission from collisions o f heavy ions and its relation to the description o f the collision process. The presence o f particles emitted before the attainment of equilibrium was established long ago in light-ion i n d u c e d reactions 1). Based on this knowledge, it is interesting to identify and to study such a c o m p o n e n t in heavy-ion i n d u c e d reactions too. I n fact, observation o f nonequilibrium emission has been reported in a n u m b e r o f experimental particle spectra from such reactions 2-8). A clear identification of nonequilibrium c o m p o n e n t s in particle spectra is, however, seldom straightforward. In the absence of a detailed dynamical treatment o f the evolution o f the collision process, one can try to extract these c o m p o n e n t s by parametrizing particle spectra with thermallyemitting moving sources9-14). The motivation and the relative advantages and disadvantages of such a parametric d e c o m p o s i t i o n are well known. In the present p a p e r we apply a refined d e c o m p o s i t i o n m e t h o d to fit n u c l e o n spectra f r o m collisions o f nearly symmetric and asymmetric nuclear colliding systems and analyze the obtained angle-integrated spectra with a B o l t z m a n n master-equation approach. The analysis is p e r f o r m e d in two states. In the first stage (sect. 2), inclusive p r o t o n spectra are d e c o m p o s e d into c o m p o n e n t s stemming from various physically plausible moving sources. In the second stage (sects. 3 and 4), the obtained d e c o m p o s i t i o n serves as a basis for the analysis in terms o f a Boltzmann master-equation approach. For this analysis, we use the H a r p - M i l l e r - B e r n e 15) equilibration m o d e l a d a p t e d 0375-9474/88/$03.50 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)

M. Korolija et al. / Initial degrees offreedom

by Blann

16) for heavy-ion

n,, the number

of degrees

This work continues

collisions.

The most relevant

of freedom

the investigation

and on the entrance-channel

in this model

energy

the conclusions

is

of the system*.

of n, on the incident

of the dependence

mass asymmetry;

data analyzed

parameter

that share the initial

2. Data. Multisource The experimental

443

energy

are given in sect. 5.

analysis of the spectra

in this paper were obtained

at the Holifield

Heavy

Ion Research Facility of the Oak Ridge National Laboratory. Inclusive proton spectra obtained by bombarding *‘Al, 46Ti, 60Ni, ‘*‘Sn, ‘24Sn and 19’Au targets with 32S beams at incident

energies

of 503.7 and 678.8 MeV (15.7 and 21.2 MeV/nucleon)

were measured at six angles: 13= loo, 30°, 50”, 70”, 110” and 144” (lab). Details of the experiment have been published in a separate paper I’). The experimental spectra were fitted using four sources which we associate with the principal processes governing particle emission from heavy-ion collisions in the energy range considered. These are: equilibrium emission from a source having the velocity of the composite system (source l), emission from two sources simulating the binary fragments (sources 2 and 3, respectively) and from a source simulating the preequilibrium emission from the composite system (source 4). In decomposing the spectra, we assume the preponderance of the evaporation emission from the compound nucleus and the (slow) target-like fragment (sources 1 and 3) at backward angles and, at forward angles, the preponderance of the evaporation from the (fast) projectile-like fragment (source 2) and of the emission from a preequilibrium source (nonequilibrated composite system, source 4). The last source was introduced to yield the high-energy part of the spectra not accounted for by the first three sources. For protons

emitted

from the first three sources we expect an isotropic

distribution in the source systems. for the invariant cross section:

We, therefore,

($)(&)

postulate

system.

Transforming

(1)

eq. (1) into the laboratory

xexe i-C&+The transformation parameter E, giving l

Precisely

speaking,

expression

=*iE1/2e-E/rz,

where Ai and 7; (i = 1,2,3) are, respectively, the normalization temperature of the emitting sources and E is the emitted-nucleon the source

maxwellian

the following

2=

constant and the kinetic energy in

system yields

COS @lab+ Ei)/

z]

.

from the source to the laboratory system is governed the per nucleon energy of the source in the laboratory the initial

number

of degrees

of freedom

is no- 1

(2)

by the frame.

444

M. Korolija et al. / Initial degrees of freedom

For source 1 (simulating compound-nucleus emission), the parameter e = el was calculated from the center-of-mass motion of the composite system, while A1 and T1 were free parameters. For sources 2 and 3 simulating the emission from the binary fragments, all parameters (A, e and T) were kept free and their values obtained from the fit procedure. Using the above three sources, we were able to reproduce the bulk of the experimental spectra at all angles. Discrepancies arose for the highest emission energies at forward angles. To account for the missing yield, we introduced an additional source associated with the preequilibrium emission from the composite system. For this source, we assumed that the emission is governed by the expression

p / k d E d O / = A4E1/2e-E/T4"

(3)

This expression introduces a forward-peaked asymmetry already in the emittingsource system. The degree of anisotropy, given by the parameter A0, depends on the energy of the emitted particles through the relation R. zlO >>-2~r/K, with K the nucleon wave number and R the radius of the composite system. Such an ansatz was used in light-ion induced reactions by Mantzouranis et al. 18) and by Blann 19) when calculating the linear momentum transfer in nucleus-nucleus collisions; in ref 20) we used it for the first time in analyzing the spectra and angular distributions of nucleons from heavy-ion-induced reactions. The transformation of eq. (3) into the laboratory system yields

lab

x exp [--(Elab-- 2 ~

COS 0lab+ e4)/T4]

x exp ( --(RcN/28.72)(Elab- 2X/~vdbe4 COS 0lab+ ~4) 1/2

[ × arc cos

%/~'ab COS 0lab -- N~4

t(E,ab--~

II

~OS0---,~+~)'/~J J

(5)

with A4 and T4 as free parameters obtained from fitting the unaccounted high-energy component of the forward-angle spectra. In calculating the angular parameter zl0, defined previously, we took R as the compound-nucleus radius RCN. The factor 1/28.72 in eq. (4) comes from expressing all energies in MeV and RcN in fm. Obviously, e4 = el. While expressions (2) and (4) were derived for neutron emission, their validity for describing proton emission could be achieved by replacing Elab by Elab--Vcai for all sources, where Vcni is the proton-source Coulomb barrier (we assume no emission for Eta b < VCBi).

TABLE 1

197Au

124812

27A1 46Ti 6°Ni 12°Sn

0.031 0.053 0.041 0.070 0.025 0.043

0.035 0.058 0.051 0.051 0.027 0.038

27A1 46Ti 6°Ni 12°Sn t24Sn

197Au

A1

Target

6.2 3.6 2.6 0.94 0.90 0.41

4.6 2.6 1.9 0.70 0.66 0.31

nucleon

(MeV)/

E1

5.73 5.50 5.88 3.96 4.29 3.13

5.04 4.81 4.62 3.47 3.41 3.09

T~ (MeV)

5.4 6.7 7.4 9.74 9.66 12.53

5.4 6.7 7.4 9.74 9.66 12.53

Vcm (MeV)

0.225 0.242 0.241 0.131 0.115 0.142

0.265 0.257 0.233 0.135 0.134 0.157

A2

13.7 12.2 13.7 13.94 12.83 10.19

10.1 11.8 11.7 7.69 7.08 9,00

(MeV)/ nucleon

~2

VcB2 (MeV) A3

3.43 3.43 3.43 3.43 3.43 3.43

0.025 0,110 0.100 0.039 0,031 0.026

3.8 3.9 3.8 4.03 4.08 4.26

3.43 3.43 3.43 3.43 3.43 3.43

0.028 0.054 0,051 0.026 0.054 0.018

Ein c = 678.8 MeV

3.2 3.1 3.2 4.33 4.23 3.67

E~,,¢ = 503.7 MeV

T2 (MeV)

1.1 0.5 0.8 0.31 0.30 0.11

1.9 1.3 0.8 0.18 0.18 0.10

(MeV)/ nucleon

E3

5,4 4,0 4,0 3.08 3.07 3.02

5.0 3,1 3.1 3.08 2.86 2.53

T3 (MeV)

2.9 4.4 5.2 7.84 7.77 10.85

2.9 4.4 5.2 7.84 7.77 10.85

VcB3 (MeV)

0.0045 0.0080 0.0660 0.031 0.019 0.046

0.0058 0.0117 0.0112 0.024 0.024 0.028

A4

6.2 3.6 2.6 0.94 0.90 0.41

4.6 2.6 1.9 0.70 0.66 0.31

nucleon

(MeV)/

E4

9.4 11.0 12.01 10.18 10.88 10.00

7.8 8.1 8.4 7.48 7.49 7.09

7"4 (MeV)

5.4 6.7 7.4 9.74 9.66 12.53

5.4 6.7 7.4 9.74 9.66 12.53

VcB4 (MeV)

Best-fit and imposed values of the parameters of the four sources used for fitting the proton spectra from the 32S+27A1, 325+46Ti, 325+6°Ni, 325+ 12°5n, 325-b 124Sn and 32S + 197Au systems at •inc = 503.7 and 678.8 MeV, respectively

4~

-~

?

~" ~

~"

M. Korolija et aL / Initial degrees offreedom

446

' -"

tU

_o

I

s.Ai-x.p

."~"~..%~ 0 r~-.~ " t

:-.~

"~,

Ein¢ = S03.7 MeV }Expdota

•

F".,,I x

~-

-3

-5

F

....

-.... ~--~-----~--~ \~ 1N, ~" ; ~s0*

=

[.

,~

t - ~

/

144" Nt 10"

t70

1

S*Ti~X*p Einc = 678.8 MeV t Exp, data

•".

".

0 - I

07

"----.1 "=-'=--..~ -. ~'~',,,.

~ ,

,.\

° • o

X*~,~xIO "~-.

,

-5

", SO*

__O

14/~* 1

"~110" I

!° • "

,.~ in

0

. . . . . "'..~, • .~

,

'~ b

S + Ni ~ X * p

Einc =503.TMeV t Exp. data

..~ . j x 1 0

, "

\.

-2 ,~

"1o UJ

=10"

-3 . . . . .

'~

70*

30* 110"

tm o

50"

144" 20

40

60

80

100

120

Eta b ( M e V )

Fig. 1. Typical best fits to experimental p r o t o n spectra: nearly symmetric systems.

M. Korol~ja et al. / Initial degrees of freedom I

,.~ E o

N

o:o -2

_~

°-%/

S J 2~,Sn ~ X+p Einc = 5033 MeV Exp. data

xll

v'%i ~°'o1 ~'o, o

~oOo~ ~, ~ , o.o o,o

-o

,O,o2-,i

?.%

°N%

o~o

"o

\

=

Of, o

D

.~

I

-6

0=40°

i

50"

30 °

110 14/-,"

+;26Sn ~ X + p

~-o

"%0

/x 10

"o

Einc =6"/B.8 MeV Exp. data

-°,,%

~o'°-O,o'°.o. ,3

~

-o UJ •~

%, .%.

R "°.o"°"

E

~%,o._ `%

\~0 .Ic

-o,

o,,~,~ a0'

%.,.

,.

11c

-S 144"

0

0

-- ~ ~ ~°o oo~'~ °

S *Au ~X,.- p Einc=503.7 MeV

-O.o %o

.~ -~ =

~!o N',~,°

-s

~

~"

o)

144"

-6

I,~

"~i ~ 30" ~0" "5 so.

2 -7

20

40

60 80 Etab ( MeV )

100

Fig. 2. Typical best fits to experimental proton spectra: asymmetric systems.

447

448

M. Korolija et aL / Initial degreesoffreedom

All the parameters used in the analysis, together with the calculated particle-source Coulomb-barrier values, are collected in table 1. Figs. 1 and 2 show the quality of the obtained fits.

3. Angle-integrated spectra Figs. 3 and 4 show the angle-integrated spectra obtained using the multisource analysis described in sect. 2 for, respectively, the (nearly symmetric) S2S + 46Ti and the (asymmetric) 32S + 124Sn reactions at 678.8 MeV. The figures display the contributions from the various sources as well as their sum. The decomposition shows that the (fast) projectile-like fragment accounts for most of the protons emitted with E~ab> 20 MeV. The angle-integrated equilibrium spectrum runs essentially parallel to that for the fast fragment, but has lower absolute values. Proton emission from the slow fragment contributes only to the low-energy part of the spectrum. Preequilibrium emission, because of its characteristic high temperature, dominates the highenergy part of the spectrum. These features are common to all analyzed systems. Table 1 shows that the fit temperatures 7"2 of the fragments are close to the fit temperatures 7"1 of the corresponding compound nuclei; such an observation was recently reported in an analysis of a coincident measurement of light particles from 14N+ 197Au [ref. 21)]. Table 2, in turn, compares the best-fit temperatures 7"1 with the statistical compound-nucleus temperatures TcN= ( E * / a ) w2, calculated from

I

S *Ti

1

Ein c : 678.8 MeV

~X*p

I

> :E

...

.,Q

I

E i

w

I

0

"\

J i

ix,. !

;\ {:~ o

-I \ -2 2O

40

!\ "~.

t 60

''..... [

I\\,'%..] 80

I00

Elab ( M e V }

Fig. 3. Angle-integratedproton spectra from the equilibrium (dashed line), preequilibrium (thin solid line), projectile-like(dotted line) and target-like (dash-dotted line) sources for the 32S+46Ti reaction at E i n c = 678.8 MeV. The thick solid line is the sum of the four components.

M. Korolo'a et aL / Initial degrees offreedom

449

J i

_,o Q)

S *12t'Sn ~

--

2

X÷ p

Einc: 678.SMeV

i i

E

1

i

v A

I.IJ "0

0

I i

\

\ \

"0 {:7) 0

i

\

'\

-1

"..

\1\.\ \k

-2

20

I

"'". ". ~

\

~o

"•.. 6O

80

]

100

Etab (MeV) Fig. 4. Same as fig. 3 for 32S+124Sn.

TABLE 2 Best-fit temperatures T l of source 1 (evaporation) and statistical c o m p o u n d nucleus temperatures TCN rget

27A1

46Ti

6ONi

120Sn

124Sn

197Au

3.47 4.30

3.41 4.28

3.09 3.44

3.96 5.07

4.29 5.04

3.13 4.13

El. c = 503.7 MeV Tl

TeN

5.04 5.81

4.81 5.53

4.62 5.29

E~.¢ = 678.8 MeV T1 TCN

5.73 6.68

5.50 6.42

5.88 6.16

the available excitation energy E * = E + Qfus and the level density parameter a set equal to ~A. The fit temperatures are close to the calculated ones; the small difference (T~ generally somewhat smaller than TCN) reflects the presence of preequilibrium emission. The rotational energy of the system will also diminish its temperature. Fig. 5 compares the preequilibrium fit temperatures TpE = T4 from the refined multisource analysis performed in the present work with (i) preequilibrium fit temperatures obtained by Holub et al. 5,22) by analyzing exclusive neutron data and ....

450

M. Korolija et al. / Initial degrees of freedom

p PRESENT A~ALYSIS '

+ S*A[ S*Ti o

o

/ l / • .,~

S*Ni

/,

• S+12OSn

10

I

• S+12~Sn

x e~

1

o ~E

5

/0:.~

[] n-ER

"

.i ~''

0

,

0

• p INCL.

I

10

I

I

20

(E-VcB) lAp (MeV/nucl.eon) Fig. 5. Preequilibrium temperature parameters TpE versus the laboratory energy per nucleon above the entrance-channel Coulomb barrier, (E - Vca)/A P. Open squares denote values of TpE extracted from neutron spectra measured in coincidence with evaporation residues [refs. 5,22)]; full squares denote TpE extracted from inclusive proton data by other authors [refs. 3,4)]. All other signs denote the temperature parameters TpE = T4 obtained in the present work from the inclusive proton spectra using the refined multisource analysis. The lines are drawn to clearly show two different trends: the solid line for TpE values obtained from exclusive data [refs. 5,22)] and the dashed line from inclusive data [refs. 3,4)]. Most of the presently obtained values of T4 lie within 15% of the solid line.

(ii) preequilibrium fit temperatures obtained by other authors 3,4) from the analysis of inclusive proton spectra by the usual moving-source method. As seen in the figure, values of preequilibrium fit temperatures T4 from the present analysis follow the absolute values and the energy trend of the temperatures obtained from the exclusive neutron data. They clearly differ from the temperatures obtained using the usual analysis of inclusive spectra. This fact strengthens the confidence in the way (refined multisource analysis) the preequilibrium component has been extracted from inclusive spectra. 4. Determination of the initial number of degrees of freedom

Our next step is to evaluate no, the number of degrees of freedom that share the initial energy. This we do by comparing the results of the Boltzmann master-equation calculation 16), containing this parameter, with angle-integrated spectra. The calculation was performed using the code* RELAX 23) primarily aimed to calculate the * It was kindly supplied to us by M. Blann.

M. Korolija et al. / Initial degrees of freedom

451

emission of particles from central collisions. Hence, the comparison is made with that part of the experimental proton spectra which comes from central collisions. This part is supplied by the yield of the two sources associated with the compoundnucleus and preequilibrium emission, respectively. The comparison procedure was described in detail elsewhere ~6,24); here we give only the outline. (i) Time evolution. The master-equation approach, contained in the model, permits to follow the time evolution of the colliding system. The model thus yields both preequilibrium and equilibrium emissions. The way of limiting the model to calculate only the high-energy component of the spectrum (containing, essentially, the preequilibrium contribution) is to terminate the time evolution when the emission of high-energy particles becomes negligible. The time integration is performed in discrete steps of 2.1 x 10 -23 S each; this interval corresponds to the average time between two subsequent nucleon-nucleon collisions in nuclear matter. Previous analysis 24) has shown that the number of steps required to account for the highenergy-component emission is between 15 and 25, regardless of the colliding system and incident energy. This yields the equilibration time to be less than 3 - 4 x 10 -22 s. Accordingly, in the present calculation we choose to terminate the calculations for all colliding systems at 18 steps. (ii) Transition rates. The evolution of the colliding system is governed by the transition rates which we calculate from nucleon-nucleon collisions in nuclear matter. The model allows for a scaling factor k of the transition rates. A value of k < 1 physically means a nucleon mean free path larger than the one obtained from nuclear matter calculations. Previous work 22,24) has shown that the value k = ¼ for the scaling factor yields the best overall fit for a variety of projectile-target combinations. This value corresponds to an increased mean free path and reduced two-body transition rates. The scaling parameter k, however, influences only the absolute values of the calculated spectra, not their slopes; hence the values of no extracted by comparing data and calculations are rather insensitive to its choice. Having thus fixed the integration time ( - 4 x 10 -22 s) and the scaling factor in the transition rates (k = ~), the only fitting parameter in the analysis was no, the intial number of degrees of freedom that share the energy. This parameter is defined by the equation 16) N,(U)AU=nl

[u°o

E * o-'

11j ,

(5)

where n~ is the number of nucleons of type i (neutrons or protons) entering the process of dissipation in the tth time interval, and N i ( U ) A U is the number of nucleons of type i in the energy interval A U centered around the excitation energy U; E* is the energy of the composite system. It can be shown 25) that, in the limit A U--> 0, no is equal to the number of "excitons" obtained from the slope analysis of the so-called Griffin plots 26).

452

M. Korolija et al. / Initial degreesoffreedom

The comparison of the summed angle-integrated spectra from sources 1 and 4 (emission following central collisions) with the values calculated using the code R E L A X yields the value no, as this is the only free parameter in the calculation. Figs. 6 and 7 show typical angle-integrated central-collision spectra compared with spectra calculated with three different values of no. This comparison gives an idea of the sensitivity of the calculation to no. Best-fit values of no are collected in table 3. We were able to fit the angle-integrated spectra from central-like collisions for a specific system using the same value of no for the two different incident energies studied. The relatively small energy difference, however, does not allow to generalize this conclusion. An energy-independent no was advocated by Blann 27); his recent analysis 28) of coincidence data from ref. 5) as well as the analysis by Fahli et al 29) support this conjecture. In fig. 8 we have collected all the values of no obtained by analyzing the experimental data for 32S induced reactions and plotted them against the entrance-channelasymmetry parameter ( A p - A T ) / ( A p + AT), where Ap and AT are the projectile and target mass numbers, respectively. Our results show an almost linear dependence of n o - A p on ( A p - A T ) / ( A p + A T ) . The figure also includes a value of no obtained

I ~ . "~;e~..~ Ei~c =678.8MqN ~•'..T.;~. ""~--,,;r.-. ;,,. .~. ..,.% / ..--o''o~ ~ . ~,',

A

> G)

~"

S . A t --,.- x + p

0

"-,<,.

Data ...... no = 20 n,= 23 ..... n,= 26

'

E LU

.....

"

"1:3 c~ o

-2

-3 r

N

-4

20

40

"~.

\

i ,i

60 Elob

80

•

\

""

xV~

100

(MoV)

Fig. 6. Results for the central-like componentof the angle-integrated proton spectra from the 325-1-27A1-'> X + p reaction at two different incident energies. Full circles show values of the spectrum obtained by summing the evaporation+ preequilibrium components. Lines show calculations using the Boltzmann master-equation approach; the solid line is the best-fit calculation; the dashed and dot-dashed lines are calculations with values of no differing from the best-fit values by ~3, respectively.

M. Korolija et aL / Initial degrees of freedom

453

3

I S + Au~X+p

E inc = 678.8 M e V

• Data

2

....

n . = 3/, n, = 37 no = 40

xlO

¢'1

E

Einc = 5 0 3 . 7 M e V

\

ILl n~ ~o -1 o

\ -2

-3 20

40

60

80

100

Etab (HEY) Fig. 7. Same as fig. 6 for 325+197Au.

TABLE 3 Best-fit values of no obtained from the analysis Target

503.7 678.8

no no

27A1

46Ti

6°Ni

23 23

28 28

29 29

12OSn |24Sn

35 35

35 35

197Au

37 37

by the analysis of coincidence proton spectra from the central-like collisions (fusionfission) of 14Nq-197Au at 490 MeV [data from ref. 21)]. This value fits well into the observed trend.

5. Conclusions Inclusive proton spectra from heavy-ion collisions induced by 325 projectiles at 503.7 and 678.8 MeV (15.7 and 21.2 MeV/nucleon) on 27A1, 46Ti, 6°Ni, 12°Sn, 124Sn and 197Au targets were analyzed in terms of a multisource parametrization. A decomposition into four sources was adopted; the sources simulated, respectively, the emission from the compound nucleus, from the binary fragments and from the

454

M. Korolija et al. / Initial degrees of freedom

I

10 + B

~ m

A

S,'At S'.-Ti

0 S*Ni •

S.,.I~Sn-

•

S.12';Sn

X S÷Au B N*Au

t,

ii

I

....

O iA -5

+ -10 - 0.8

- 0.6

- 0.&

Ap-

- 0.2

0.0

0.2

AT

A p .,. A T

Fig. 8. no-Ap versus the entrance-channel asymmetry parameter for 32S as a projectile. Equal values of no are obtained for both the incident energies analyzed. For comparison, the value of no obtained by an analysis of central-like coincidence data of taN+ 197Au at 490 MeV [ref. 21)] is also added. nonequilibrated composite system (preequilibrium source). The best-fit parameters obtained for the sources concur with this physical picture. Evidence for preequilibrium emission has been found in the high-energy part of the spectra. This evidence is based on the anisotropy of the angular distributions and high effective temperatures needed in the analysis, similar to those derived from exclusive data 5.22). The four sources, described in sect. 2, were sufficient to produce high-quality fits to the data (see figs. 1 and 2). There were, thus, no grounds for introducing an intermediate-velocity source, as suggested by several earlier analyses 3,1,,~4). Such a source was indeed introduced to account for the forward-peaked, high-energy part of the spectra; in the present analysis it is successfully supplanted by the anisotropic preequilibrium source. The angle-integrated spectra from central-like collisions (compound-nucleus and preequilibrium sources) were compared with a Boltzmann master-equation calculation performed using the code RELAX. This comparison yields (i) the equilibration time in agreement with earlier results ( - 3 - 4 x 10 -22 s) and (ii) no, the initial n u m b e r of degrees of freedom that share the total energy of the system. We have found no evidence for incident-energy dependence of no for a given colliding pair, at least in the energy range investigated. Rather, a smooth dependence

455

M. Korolija et aL / Initial degrees of freedom

o f no o n t h e e n t r a n c e - c h a n n e l s y m m e t r y w a s o b s e r v e d (see fig. 8). T h i s b e h a v i o r is a n o v e l f e a t u r e in o u r k n o w l e d g e o f t h e e a r l y s t a g e s o f h e a v y - i o n c o l l i s i o n s .

This paper

is p a r t o f a s t u d y u n d e r t h e U S - Y u g o s l a v

J F P 554 D O E / I R B .

collaboration project

It w a s i n i t i a t e d d u r i n g t h e stay o f o n e o f t h e a u t h o r s ( N . C . ) at

the Joint Institute for Heavy Ion Research (JIHIR)

at O a k R i d g e . T h e f i n a n c i a l

supports of the Joint US-Yugoslav Board for Scientific and Technological Cooperation and the JIHIR are gratefully acknowledged.

References 1) M. Blann, Ann. Rev. Nucl. Sci. 25 (1975) 125 and references therein 2) A. Gavron, J.R. Beene, R.L. Ferguson, F.E. Obenshain, F. Plasil, G.R. Young, G.A. Petit, K. Geoffroy Young, M. JS~iskelfiinen, D.G. Sarantites and C.F. Maguire, Phys. Rev. C24 (1981) 2048 3) T.C. Awes, S. Saini, G. Poggi, C.K. Gelbke, D. Cba, R. Legrain and G.D. Westfall, Phys. Rev. C25 (1982) 2361 4) R. Glasow, G. Gaul, B. Ludewigt, R. Santo, H. Ho, W. Kfihn, U. Lynen and W.F.J. Mfiller, Phys. Lett. B120 (1983) 71 5) E. Holub, D. Hilscher, (3. Ingold, U. Jahnke, H. Orf and H. Rossner, Phys. Rev. C28 (1983) 252 6) R.L. Robinson, R.L. Auble, J.B. Ball, F.E. Bertrand, R.L Ferguson, C.B. Fulmer, J. Gomez del Campo, I.Y. Lee, R.W. Novotny, G.R. Young, J.C. Wells, C.F. Maguire and H. Yamada, Phys. Rev. C27 (1983) 3006 7) M.B. Tsang, C.B. Chitwood, D.J. Fields, C.K. Gelbke, D.R. Klesch, W.G. Lynch, K. Kwiatkowski and V.E. Viola, Jr., Phys. Rev. Lett. 52 (1984) 1967 8) H. Machner, (3. Riepe, D. Proti6, H.G. Bohlen and H. Fuchs, Phys. Rev. C30 (1985) 443 9) T.J.M. Symons, P. Doll, M. Bini, D.L. Hendrie, J. Mahoney, (3. Mantzouranis, D.K. Scott, K. Van Bibber, Y.P. Viyogi, H.H. Wieman and C.K. Gelbke, Phys. Lett. B94 (1980) 131 10) D.K. Scott, Nucl. Phys. A354 (1981) 375 11) G.D. Westfall, B.V. Jacak, N. Anantaraman, M.W. Curtin, G.M. Crawley, C.K. Gelbke, B. Hasselquist, W.G. Lynch, D.K. Scott, M.B. Tsang, M.J. Murphy, T.J.M. Symons, R. Legrain and T.J. Majors, Phys. Lett. Bl16 (1982) 118 12) R.L. Auble, J.B. Ball, F.E. Bertrand, C.B. Fulmer, D.C. Hensley, I.Y. Lee, R.L. Robinson, P.H. Stelson, D.L. Hendrie, H.D. Holmgren, J.D. Silk and H. Breuer, Phys. Rev. Lett. 49 (1982) 441 13) G. Caskey, A. Galonsky, B. Remington, M.B. Tsang, C.K. Gelbke, A. Kiss, F. Deak, Z. Seres, J.J. Kolata, J. Hinnefeld and J. Kasagi, Phys. Rev. C31 (1985) 1597 14) C.K. Gelbke, Nucl. Phys. A400 (1983) 473 15) G.D. Harp, J.M. Miller and B.J. Berne, Phys. Rev. 165 (1968) 1166 16) M. Blann, Phys. Rev. C23 (1981) 205 17) R.L. Auble, J.B. Ball, F.E. Bertrand, I.Y. Lee, R.L. Robinson and G.R. Young, Phys. Rev. C37 (1988) 390 18) G. Mantzouranis, H.A. Weidenmfiller and D. Agassi, Z. Phys. A276 (1976) 145 19) M. Blann, Phys. Rev. C31 (1985) 295 20) M. Korolija, N. Cindro, R. (~aplar, R.L. Auble, J.B. Ball and R.L. Robinson, Z. Phys. A327 (1987) 237 21) Z. Chen, C.K. Gelbke, J. Pochodzalla, C.B. Chitwood, D.J. Fields, W.G. Gong, W.G. Lynch and M.B. Tsang, MSUCL-report no. 596, 1987 22) E. Holub, D. Hilscher, G. Ingold, U. Jahnke, H. Orf, H. Rossner, W.P. Zank, W.U. Schr6der, H. Gemmeke, K. Keller, L. Lassen and W. Lficking, Phys. Rev. C33 (1986) 143 23) M. Blann and G.D. Harp, code RELAX 24) E. Holub, M. Korolija and N. Cindro, Z. Phys. A314 (1983) 347

456

M. Korolija et al. / Initial degrees of freedom

25) N. Cindro, M. Korolija and E. Holub, in Fundamental problems of heavy ion collisions, ed. N. Cindro, W. Greiner and R. Caplar (World Scientific, Singapore, 1984) p. 301 26) J.J. Griffin, Phys. Lett. B24 (1967) 5 27) M. Blann, in Proc. Int. School on nuclear physics, 1974, ed. A. Ciocanel (Editura Academiei RS Romaniei, Bucharest, 1976) p. 249; Nucl. Phys. A235 (2974) 211 28) M. Blann, LLNL report UCRL 94313, 1983; in Proc. Symp. on the many facets of heavy ion fusion, ANL-PHY-86-1, 1986; Phys. Rev. C31 (1985) 1245 29) A. Fahli, J. P. Coffin, G. Guillaume, B. Heusch, F. Jundt, F. Rami, P. Wagner, P. Fintz, A.J. Cole, S. Kox, Y. Schutz and N. Cindro, Z. Phys. A326 (1987) 169