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Do nuclei flow at high energies?
Volume 110B, number 3,4
PHYSICS LETTERS
1 April 1982
DO N U C L H FLOW AT HIGH ENERGIES ? ¢:
M. GYULASSY, K.A. FRANKEL and H. STOCKER Nuclear Science Division, Lawrence Berkeley Laboratory, University of California, Berkeley, CA 94720, USA Received 29 January 1982
Nuclear flow patterns for Ela b = 250 MeV/nucleon are generated using the cascade code of Cugnon et al. and analyzed in terms of a kinetic flow tensor. The effect of increasing the nucleon-nucleon cross section and the difference between repulsive and stochastic scattering styles are studied. A flow diagram is proposed to aid the experimental search for such flow patterns.
An important motivation for studying nuclear collisions at energies ~1 GeV/nucleon is the hope that the nuclear matter equation of state, W(p, r), could be deduced experimentally. However, the program to extract W(p, r) from nuclear collision data relies on the assumption that hydrodynamics provides an adequate dynamical framework at such energies [1 ]. Until recently, experimental tests of hydrodynamic behavior have focused on inclusive data [2]. The results indicate that such data are rather insensitive to dynamical details and are consistent with both fluid [3,4] and cascade [5,6] behavior. New data [7] on central triggered Ne + U have provided, on the other hand, a more stringent test of dynamical models. The qualitative agreement of recent hydrodynamic plus evaporation calculations with the features of those data provides at this time the strongest indication that hydrodynamic flow may occur in nuclear collisions [8]. However, because these calculations fail to reproduce the high-energy (E > 50 MeV/nucleon) fragment spectra and because cascade calculations have thus far ignored composite formation, no definitive conclusions can be drawn from the present data. In order to test possible hydrodynamic behavior more directly, it has been proposed that exclusive variables such as the longitudinal energy fraction [9], bounce angle [10], thrust [11] and sphericity [12] should be measured. With such exclusive variables it is easy to differentiate between gas and fluid behavior. The former [5] leads to collective flow only near the
beam direction, while fluid dynamics [10] predicts significant collective flow at finite angles. To understand better this qualitative difference between cascade and fluid results we investigate in this letter the transition from transparency to thermalization and finally to hydrodynamic flow as a function of particle number, A, effective nucleon-nucleon cross section, Oeff, and scattering style [13]. The cascade program developed by Cugnon et al. [5] is used to generate exclusive "data" as a function of A and Oeff. We analyz e the flow patterns by constructing a kinetic flow tensor event by event. This tensor is the natural generalization of sphericity [12,14] to reactions in which composite fragments are produced. To associate a flow pattern or an event shape with a set of measured final momenta (p(v):v = 1,N} the simplest procedure is to construct a weighted flow tensor: N
Fij = ~
u=l
WvPi(v) pj(v) ,
(1)
where wv is the weight given to fragment v. The familiar sphericity tensor [12,14] follows when w~ is a constant, independent of v. However, for nuclear collisions sphericity is inadequate since it gives the incorrect weight to composite fragments [11 [. To achieve the correct weighting of composites while retaining the simple analytic properties of sphericity, we propose to use wv = (2mu)-t, where my is the 185
Volume ll0B, number 3,4
PHYSICS LETTERS
mass of the uth fragment. With this weight, eq. (1) defines the kinetic flow tensor. In terms of the eigenvalues fn and orthonormal eigen (column) vectors en ofF.
F = f l e l e ' { + f2e2e2++ f3e3e3, +
(2)
In diagonal form F specifies an ellipsoid in momentum space with principal axes along en and radii V~n. A cigar pattern, which is oriented along the z axis, would lead tOll > f 2 =f3 and e I =2, e2 = 2 , e 3 = p . A pancake pattern would have f l < f 2 =f3. The eigenvalues satisfy the cubic equation f3n + a2f2n + alfn + a0 = 0, with a 2 = - ( F l l + F22 + F33 ) ,
(3)
a 1 = F l l F 2 2 + F11F33 + F22F33 - F 2 2 - F 2 3 - F23,
(4)
a 0 = F l l F 2 3 + F22F23 + F33F22 - F 11F22F33 - 2F12F 13F23 ,
(5)
The three real solutions (n = 1,2, 3) can be compactly written as
f n -- - 51a 2 + 2p cos[~cos-l(r/p 3) + (n - 1) ~rr],
(6)
in terms of standard auxiliary variables P = ](a22 - 3al)1/2 ,
(7)
r = (ala 2 - 3a0)/6 - a 3 / 2 7 .
(8)
The eigenvectors are most conveniently evaluated in polar coordinates [e+ = (sin On cos ~bn, sin On sin q~n, cos On)] with polar and azimuthal angles given by cos On = [1 + C2n+ (fn - F33 - F23Cn)2/F213]-l/2, (9) tan Cn = cnF13/(fn - F33 - F23cn) , with Cn = (en "P)/(en " 2) given by (Ell - fn)(f33 - fn) - F23 Cn = F12F13 _ F23(F11 - f n )
(10)
The three orthogonal flow vectors that specify the event shape can now be defined in polar coordinates to be.fn = (fn, On, C~n). We emphasize that event shape analysis makes sense only in the nucleus-nucleus center of mass system since the eigenvalues o f F specify an ellipsoid centered at the origin. 186
1 April 1982
Note that the longitudinal energy fraction proposed in ref. [9] is simply F33/tr F. However, the kinetic flow tensor contains substantially more information (six independent numbers) than F33 and has a simpler, more intuitive interpretation than the Legendre coefficients of the angular pattern [9]. On the other hand, the Legendre coefficients of the event in the rotated coordinate system spanned by the e n would be useful for describing pear shape events for asymmetric reactions such as Ne + U. In order to display the range of flow patterns appropriate for a given reaction we propose the following flow diagram. First order the flow eigenvalues such that f l >I'2 >]'3 so that 01 is the polar angle of the maximum kinetic flow. We will call 01 the flow angle OF. The flow diagram is obtained by plotting OF versus the kinetic flow ratios f i l l 3 a n d f 2 / f 3 on the same plot. For a fixed OF, a sphere corresponds to f l / f 3 =f2/f3 = 1, while a cigar with major axis rotated by OF corresponds tOll/f3 > f 2 / f 3 = 1. A squashed cigar h a s f l / f 3 > f 2 / f 3 > 1. A pancake event with a normal oriented along the beam axis corresponds to a flow angle OF = 90 ° a n d f l / f 3 =f2/f3 > 1. An important experimental advantage of the flow diagram is that the ratiosfl/f3,f2/f3 are insensitive to missed particle such as neutrons. In fig. 1, we show the flow plot for Ne + Ne, Ar + Ar, and U + U at Ela b = 250 MeV/nucleon. This energy was chosen to maximize sensitivity to dynamical details [ 15]. Indicated along the boundaries are typical event shapes and orientations defined by a given OF and flow ratio. For this plot the free-space cross sections and standard stochastic NN scattering style (ONNchosen at random from dONN/df2cm,~bNN chosen randomly between 0 and 2rr at the distance of closest approach) were used in the cascade calculation. For Ne and Ar, 50 events were generated at each impact parameter, b/bmax = 0, 0.1 ..... 0.9. For U, 10 events per b were generated. It is important to emphasize that in contrast to differential cross section, event shapes can be determined from a relatively small number of events. The average values (OF)b,(fl/f3)b, (f2/f3)b as a function of impact parameter are indicated by points along the curves in units of bmax/10. Since the probability that the impact parameter lies between b and b + db is proportional to b, the numbers beside the symbols also indicate the relative intensity of such event shapes. Thus, standard cascade [5] predicts that there are five times more events with
Volume 110B, number 3,4
PHYSICS LETTERS
@
\ 80.n
'0 70.0 60.0
•
--
N e -I- N e
--
Ar-I-
Ar
.--u+u
"z t l 0 . 0 LI , /
0
.^
, 2?0
.?0
KIN.,C
s.'0
<
.Lo...,0s
1 April 1982
fixed b. Thus, Ar is also m u c h more transparent than U. It is nevertheless clear f r o m fig. 1 that there is a continuous e v o l u t i o n o f event shapes as A increases to smaller aspect ratios and larger flow angles. The most interesting question is w h e t h e r U + U behaves as a nonviscous fluid. The answer using the standard cascade is no. We can determine one point on the flow diagram f r o m existing h y d r o d y n a m i c calculations. F r o m ref. [9], F 3 3 / t r F ~ 0.17 for b = 0 corresponding to OF = 9 0 ° , f l / f 3 = f 2 / f 3 ~ 2.4 as shown by the arrow in fig. 1. In contrast, the cascade produces near isotropy for U + U. Therefore, we find a continuous transition f r o m transparency to the isotropic thermalization as a f u n c t i o n o f A . However, the standard intranuclear cascade [5] does n o t converge to the fluid limit. Calculations w i t h A = 500, 1000, 2 0 0 0 confirm this conclusion. Besides increasing A we have tried using larger effective NN cross sections to induce fluid behavior. In fig. 2 the results for U + U using c%ff = 3ONN are shown by the curves $3. Such large Oeff could arise,
Fig. 1. Flow diagram showing the average polar angle of the maximum kinetic flow versus average flow ratios f i l l 3 (solid) and f2/f3 (dashed). Symmatric A + A collisions "data" at 250 MeV/nucleon were computed via the cascade code of Cugnon et al. [5]. The numbers beside the symbols refer to the value of b/bma x in steps of 0.1. The arrow locates the nonviscous hydrodynamic flow pattern [9] for b = 0.
OF ~ 10 ° a n d f l / f 3 ~ 5 than w i t h OF ~ 25 ° and
f l / f 3 "~ 2.5 for Ne + Ne. F o r all A , the largest 0 F occurs for b = 0. However, the flow characteristics are strongly A dependent. F o r b = 0, OF = 26-+ 20, 40-+ 25, 46-+ 28, 77-+ 9, and f l / f 3 = 2.3 -+ 0.6, 1.7 + 0.3, 1.4 + 0.1, 1.2 -+ 0.1 for A = 20, 40, 100 ( n o t plotted), 238, respectively. The -+A numbers indicate the rms fluctuations around the mean values due to finite particle n u m b e r effects. The numerical uncertainties o f the m e a n values is estimated to be A/X/~, where n is the n u m b e r o f events analyzed. In terms o f a c o n t o u r or scatter flow plot the average values define a ridge and A indicates the thickness o f that ridge. The gap between f i l l 3 ~ 2.3 a n d f 2 / f 3 ~ 1.5 for b = 0 Ne + Ne as well as the large rms fluctuations indicate that Ne is t o o small to therrealize c o m p l e t e l y . Even though the Ar ridge and U ridge coincide for b/bma x > 0.2, n o t e that the points are out o f phase w i t h f l / f 3 ( A r ) > f l / f 3 ( U ) for each
g ..J
KINETIC FLON BRTIOS
Fig. 2. As fig. 1, for U + U at 250 MeV/nucleon as function of scattering style. $3 and P3 correspond to stochastic and eq. (11) scattering styles, respectively, with aeff/aNN = 3; P5 corresponds to eq. (11) with aeff/ONN = 5. 187
Volume 110B, number 3,4
PHYSICS LETTERS
for example, if critical scattering phenomena [16] would occur in nuclear collisions or if many-body forces become important at high densities. Although the flow angle increases for fixed b relative to those in fig. 1 with Oeff = UNN, the flow at b = 0 is still far weaker than the hydrodynamic point. Therefore, X/R ,~ 1 is not sufficient for fluid behavior. Could flow be sensitive to the details on the NN scattering style? Recall that the standard scattering style corresponds to a stochastic classical force with a random sign [13]. Therefore, the m o m e n t u m transfer, q, and the relative coordinate, r, between two nucleons at the scattering time are assumed to be completely uncorrelated. Classically, on the other hand, any potential leads to definite correlations between q and r. For example, a repulsive potential leads to
q.r>O
and
q.(r×p)=O,
(11)
where p is the incident relative momentum of the two nucleons. Eq. (11) specifies the repulsive, in-scattering plane style as considered in ref. [13]. We have performed cascade calculations imposing eq. (11). With Oeff = ONN no significant difference was found in the flow diagram between stochastic and repulsive scattering styles. However, with Oeff = 3ONN we find (comparing curves $3 with P3 in fig. 2) considerable dependence on style. Only the nonrandom style eq. (11) leads to collective flow similar to nonviscous hydrodynamic flow. As Oeff increases the flow becomes more pronounced. For Oeff = 5ONN with eq. (11) style (curve P5), the flow r a t i o f l / f 3 even exceeds the fluid value [9]. Thus, the flow obtained with nonviscous one-fluid hydrodynamics represents only one possible class of flow patterns as a function o f b . We conjecture that the variations of Oeff and scattering style considered here correspond in terms of viscous hydrodynamics to substantial variations in the transport properties of nuclei. For example, stochastic scattering ($3) may lead to larger viscosity effects than repulsive scattering (P3) thereby damping flow gradients. Also, P5 could lead to larger flow gradients than nonviscous hydrodynamics by simulating high immiscibility of projectile
188
1 April 1982
and target nuclei. These conjectures are currently under investigation. In conclusion, we have found that in a cascade model, nuclei exhibit hydrodynamic flow if the NN collisions are significantly modified [16] in dense matter. However, the possibility o f additional nuclear flow resulting from mean field dynamics (neglected in cascade) still needs to be investigated. In any case, the flow analysis proposed here could be used profitably to quantify actual nuclear flow as exclusive data become available. We are very grateful to J. Cugnon for making his cascade code available for our studies. This work was supported by the Director, Office of Energy Research, Division of Nuclear Physics of the Office of High Energy and Nuclear Physics of the US Department of Energy under Contract W-7405-ENG-48.
References [1] J.R. Nix, Prog. Part. Nucl. Phys. 2 (1979) 237. [2] A. Sandoval et al., Phys. Rev. C21 (1980) 1321. [3] J.R. Nix and D. Strottman, Phys. Rev. C23 (1981) 2548. [4] H. St6cker et al., Phys. Rev. Lett. 44 (1980) 725. [5] J. Cugnon et al., Nucl. Phys. A352 (1981) 505; Phys. Rev. C22 (1981) 2094. [6] Y. Yarn and Z. Fraenkel, Phys. Rev. C20 (1979) 2227; C24 (1981) 488. [7] R. Stock et al., Phys. Rev. Lett. 44 (1980) 1243. [8] H. St~Sckeret al., LBL-12634, preprint (1981). [9] G. Bertsch and A.A. Amsden, Phys. Rev. C18 (1978) 1293. [10] H. St~Sckeret al., LBL-11774 preprint (1980); G. Buchwald et al., Phys. Rev. C24 (1981) 135. [11] J. Kapusta and D. Strottman, Phys. Lett. 106B (1981) 33. [12] J. Knoll, GSI-81-18 preprint (1981). [13] E. Halbert, Phys. Rev. C23 (1981) 295. [14] S. Brandt and H.D. Dahmen, Part. Fields C1 (1979) 61. [15] H. St~Scker,M. Gyulassy and J. Boguta, Phys. Lett. 103B (1981) 269. [16] M. Gyulassy and W. Greiner, Ann. Phys. 109 (1977) 485.
PHYSICS LETTERS
1 April 1982
DO N U C L H FLOW AT HIGH ENERGIES ? ¢:
M. GYULASSY, K.A. FRANKEL and H. STOCKER Nuclear Science Division, Lawrence Berkeley Laboratory, University of California, Berkeley, CA 94720, USA Received 29 January 1982
Nuclear flow patterns for Ela b = 250 MeV/nucleon are generated using the cascade code of Cugnon et al. and analyzed in terms of a kinetic flow tensor. The effect of increasing the nucleon-nucleon cross section and the difference between repulsive and stochastic scattering styles are studied. A flow diagram is proposed to aid the experimental search for such flow patterns.
An important motivation for studying nuclear collisions at energies ~1 GeV/nucleon is the hope that the nuclear matter equation of state, W(p, r), could be deduced experimentally. However, the program to extract W(p, r) from nuclear collision data relies on the assumption that hydrodynamics provides an adequate dynamical framework at such energies [1 ]. Until recently, experimental tests of hydrodynamic behavior have focused on inclusive data [2]. The results indicate that such data are rather insensitive to dynamical details and are consistent with both fluid [3,4] and cascade [5,6] behavior. New data [7] on central triggered Ne + U have provided, on the other hand, a more stringent test of dynamical models. The qualitative agreement of recent hydrodynamic plus evaporation calculations with the features of those data provides at this time the strongest indication that hydrodynamic flow may occur in nuclear collisions [8]. However, because these calculations fail to reproduce the high-energy (E > 50 MeV/nucleon) fragment spectra and because cascade calculations have thus far ignored composite formation, no definitive conclusions can be drawn from the present data. In order to test possible hydrodynamic behavior more directly, it has been proposed that exclusive variables such as the longitudinal energy fraction [9], bounce angle [10], thrust [11] and sphericity [12] should be measured. With such exclusive variables it is easy to differentiate between gas and fluid behavior. The former [5] leads to collective flow only near the
beam direction, while fluid dynamics [10] predicts significant collective flow at finite angles. To understand better this qualitative difference between cascade and fluid results we investigate in this letter the transition from transparency to thermalization and finally to hydrodynamic flow as a function of particle number, A, effective nucleon-nucleon cross section, Oeff, and scattering style [13]. The cascade program developed by Cugnon et al. [5] is used to generate exclusive "data" as a function of A and Oeff. We analyz e the flow patterns by constructing a kinetic flow tensor event by event. This tensor is the natural generalization of sphericity [12,14] to reactions in which composite fragments are produced. To associate a flow pattern or an event shape with a set of measured final momenta (p(v):v = 1,N} the simplest procedure is to construct a weighted flow tensor: N
Fij = ~
u=l
WvPi(v) pj(v) ,
(1)
where wv is the weight given to fragment v. The familiar sphericity tensor [12,14] follows when w~ is a constant, independent of v. However, for nuclear collisions sphericity is inadequate since it gives the incorrect weight to composite fragments [11 [. To achieve the correct weighting of composites while retaining the simple analytic properties of sphericity, we propose to use wv = (2mu)-t, where my is the 185
Volume ll0B, number 3,4
PHYSICS LETTERS
mass of the uth fragment. With this weight, eq. (1) defines the kinetic flow tensor. In terms of the eigenvalues fn and orthonormal eigen (column) vectors en ofF.
F = f l e l e ' { + f2e2e2++ f3e3e3, +
(2)
In diagonal form F specifies an ellipsoid in momentum space with principal axes along en and radii V~n. A cigar pattern, which is oriented along the z axis, would lead tOll > f 2 =f3 and e I =2, e2 = 2 , e 3 = p . A pancake pattern would have f l < f 2 =f3. The eigenvalues satisfy the cubic equation f3n + a2f2n + alfn + a0 = 0, with a 2 = - ( F l l + F22 + F33 ) ,
(3)
a 1 = F l l F 2 2 + F11F33 + F22F33 - F 2 2 - F 2 3 - F23,
(4)
a 0 = F l l F 2 3 + F22F23 + F33F22 - F 11F22F33 - 2F12F 13F23 ,
(5)
The three real solutions (n = 1,2, 3) can be compactly written as
f n -- - 51a 2 + 2p cos[~cos-l(r/p 3) + (n - 1) ~rr],
(6)
in terms of standard auxiliary variables P = ](a22 - 3al)1/2 ,
(7)
r = (ala 2 - 3a0)/6 - a 3 / 2 7 .
(8)
The eigenvectors are most conveniently evaluated in polar coordinates [e+ = (sin On cos ~bn, sin On sin q~n, cos On)] with polar and azimuthal angles given by cos On = [1 + C2n+ (fn - F33 - F23Cn)2/F213]-l/2, (9) tan Cn = cnF13/(fn - F33 - F23cn) , with Cn = (en "P)/(en " 2) given by (Ell - fn)(f33 - fn) - F23 Cn = F12F13 _ F23(F11 - f n )
(10)
The three orthogonal flow vectors that specify the event shape can now be defined in polar coordinates to be.fn = (fn, On, C~n). We emphasize that event shape analysis makes sense only in the nucleus-nucleus center of mass system since the eigenvalues o f F specify an ellipsoid centered at the origin. 186
1 April 1982
Note that the longitudinal energy fraction proposed in ref. [9] is simply F33/tr F. However, the kinetic flow tensor contains substantially more information (six independent numbers) than F33 and has a simpler, more intuitive interpretation than the Legendre coefficients of the angular pattern [9]. On the other hand, the Legendre coefficients of the event in the rotated coordinate system spanned by the e n would be useful for describing pear shape events for asymmetric reactions such as Ne + U. In order to display the range of flow patterns appropriate for a given reaction we propose the following flow diagram. First order the flow eigenvalues such that f l >I'2 >]'3 so that 01 is the polar angle of the maximum kinetic flow. We will call 01 the flow angle OF. The flow diagram is obtained by plotting OF versus the kinetic flow ratios f i l l 3 a n d f 2 / f 3 on the same plot. For a fixed OF, a sphere corresponds to f l / f 3 =f2/f3 = 1, while a cigar with major axis rotated by OF corresponds tOll/f3 > f 2 / f 3 = 1. A squashed cigar h a s f l / f 3 > f 2 / f 3 > 1. A pancake event with a normal oriented along the beam axis corresponds to a flow angle OF = 90 ° a n d f l / f 3 =f2/f3 > 1. An important experimental advantage of the flow diagram is that the ratiosfl/f3,f2/f3 are insensitive to missed particle such as neutrons. In fig. 1, we show the flow plot for Ne + Ne, Ar + Ar, and U + U at Ela b = 250 MeV/nucleon. This energy was chosen to maximize sensitivity to dynamical details [ 15]. Indicated along the boundaries are typical event shapes and orientations defined by a given OF and flow ratio. For this plot the free-space cross sections and standard stochastic NN scattering style (ONNchosen at random from dONN/df2cm,~bNN chosen randomly between 0 and 2rr at the distance of closest approach) were used in the cascade calculation. For Ne and Ar, 50 events were generated at each impact parameter, b/bmax = 0, 0.1 ..... 0.9. For U, 10 events per b were generated. It is important to emphasize that in contrast to differential cross section, event shapes can be determined from a relatively small number of events. The average values (OF)b,(fl/f3)b, (f2/f3)b as a function of impact parameter are indicated by points along the curves in units of bmax/10. Since the probability that the impact parameter lies between b and b + db is proportional to b, the numbers beside the symbols also indicate the relative intensity of such event shapes. Thus, standard cascade [5] predicts that there are five times more events with
Volume 110B, number 3,4
PHYSICS LETTERS
@
\ 80.n
'0 70.0 60.0
•
--
N e -I- N e
--
Ar-I-
Ar
.--u+u
"z t l 0 . 0 LI , /
0
.^
, 2?0
.?0
KIN.,C
s.'0
<
.Lo...,0s
1 April 1982
fixed b. Thus, Ar is also m u c h more transparent than U. It is nevertheless clear f r o m fig. 1 that there is a continuous e v o l u t i o n o f event shapes as A increases to smaller aspect ratios and larger flow angles. The most interesting question is w h e t h e r U + U behaves as a nonviscous fluid. The answer using the standard cascade is no. We can determine one point on the flow diagram f r o m existing h y d r o d y n a m i c calculations. F r o m ref. [9], F 3 3 / t r F ~ 0.17 for b = 0 corresponding to OF = 9 0 ° , f l / f 3 = f 2 / f 3 ~ 2.4 as shown by the arrow in fig. 1. In contrast, the cascade produces near isotropy for U + U. Therefore, we find a continuous transition f r o m transparency to the isotropic thermalization as a f u n c t i o n o f A . However, the standard intranuclear cascade [5] does n o t converge to the fluid limit. Calculations w i t h A = 500, 1000, 2 0 0 0 confirm this conclusion. Besides increasing A we have tried using larger effective NN cross sections to induce fluid behavior. In fig. 2 the results for U + U using c%ff = 3ONN are shown by the curves $3. Such large Oeff could arise,
Fig. 1. Flow diagram showing the average polar angle of the maximum kinetic flow versus average flow ratios f i l l 3 (solid) and f2/f3 (dashed). Symmatric A + A collisions "data" at 250 MeV/nucleon were computed via the cascade code of Cugnon et al. [5]. The numbers beside the symbols refer to the value of b/bma x in steps of 0.1. The arrow locates the nonviscous hydrodynamic flow pattern [9] for b = 0.
OF ~ 10 ° a n d f l / f 3 ~ 5 than w i t h OF ~ 25 ° and
f l / f 3 "~ 2.5 for Ne + Ne. F o r all A , the largest 0 F occurs for b = 0. However, the flow characteristics are strongly A dependent. F o r b = 0, OF = 26-+ 20, 40-+ 25, 46-+ 28, 77-+ 9, and f l / f 3 = 2.3 -+ 0.6, 1.7 + 0.3, 1.4 + 0.1, 1.2 -+ 0.1 for A = 20, 40, 100 ( n o t plotted), 238, respectively. The -+A numbers indicate the rms fluctuations around the mean values due to finite particle n u m b e r effects. The numerical uncertainties o f the m e a n values is estimated to be A/X/~, where n is the n u m b e r o f events analyzed. In terms o f a c o n t o u r or scatter flow plot the average values define a ridge and A indicates the thickness o f that ridge. The gap between f i l l 3 ~ 2.3 a n d f 2 / f 3 ~ 1.5 for b = 0 Ne + Ne as well as the large rms fluctuations indicate that Ne is t o o small to therrealize c o m p l e t e l y . Even though the Ar ridge and U ridge coincide for b/bma x > 0.2, n o t e that the points are out o f phase w i t h f l / f 3 ( A r ) > f l / f 3 ( U ) for each
g ..J
KINETIC FLON BRTIOS
Fig. 2. As fig. 1, for U + U at 250 MeV/nucleon as function of scattering style. $3 and P3 correspond to stochastic and eq. (11) scattering styles, respectively, with aeff/aNN = 3; P5 corresponds to eq. (11) with aeff/ONN = 5. 187
Volume 110B, number 3,4
PHYSICS LETTERS
for example, if critical scattering phenomena [16] would occur in nuclear collisions or if many-body forces become important at high densities. Although the flow angle increases for fixed b relative to those in fig. 1 with Oeff = UNN, the flow at b = 0 is still far weaker than the hydrodynamic point. Therefore, X/R ,~ 1 is not sufficient for fluid behavior. Could flow be sensitive to the details on the NN scattering style? Recall that the standard scattering style corresponds to a stochastic classical force with a random sign [13]. Therefore, the m o m e n t u m transfer, q, and the relative coordinate, r, between two nucleons at the scattering time are assumed to be completely uncorrelated. Classically, on the other hand, any potential leads to definite correlations between q and r. For example, a repulsive potential leads to
q.r>O
and
q.(r×p)=O,
(11)
where p is the incident relative momentum of the two nucleons. Eq. (11) specifies the repulsive, in-scattering plane style as considered in ref. [13]. We have performed cascade calculations imposing eq. (11). With Oeff = ONN no significant difference was found in the flow diagram between stochastic and repulsive scattering styles. However, with Oeff = 3ONN we find (comparing curves $3 with P3 in fig. 2) considerable dependence on style. Only the nonrandom style eq. (11) leads to collective flow similar to nonviscous hydrodynamic flow. As Oeff increases the flow becomes more pronounced. For Oeff = 5ONN with eq. (11) style (curve P5), the flow r a t i o f l / f 3 even exceeds the fluid value [9]. Thus, the flow obtained with nonviscous one-fluid hydrodynamics represents only one possible class of flow patterns as a function o f b . We conjecture that the variations of Oeff and scattering style considered here correspond in terms of viscous hydrodynamics to substantial variations in the transport properties of nuclei. For example, stochastic scattering ($3) may lead to larger viscosity effects than repulsive scattering (P3) thereby damping flow gradients. Also, P5 could lead to larger flow gradients than nonviscous hydrodynamics by simulating high immiscibility of projectile
188
1 April 1982
and target nuclei. These conjectures are currently under investigation. In conclusion, we have found that in a cascade model, nuclei exhibit hydrodynamic flow if the NN collisions are significantly modified [16] in dense matter. However, the possibility o f additional nuclear flow resulting from mean field dynamics (neglected in cascade) still needs to be investigated. In any case, the flow analysis proposed here could be used profitably to quantify actual nuclear flow as exclusive data become available. We are very grateful to J. Cugnon for making his cascade code available for our studies. This work was supported by the Director, Office of Energy Research, Division of Nuclear Physics of the Office of High Energy and Nuclear Physics of the US Department of Energy under Contract W-7405-ENG-48.
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