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Reaction dynamics and deuteron production
Physics Letters B 297 ( 1992 ) 243-247 North-Holland
PHYSICS LETTERS B
Reaction dynamics and deuteron production M.B. Tsang, P. Danielewicz, D.R. Bowman, N. Carlin ~, C.K. Gelbke, W.G. Gong 2, Y.D. Kim 3, W.G. Lynch, L. Phair, R.T. de Souza 3 and F. Zhu National Superconducting Cyclotron Laboratory and Department of Physics and Astronomy, Michtgan State University, East Lansing, M148824, USA Received 23 April 1992; revised manuscript received 16 October 1992
Proton and deuteron azimuthal distributions with respect to the reaction plane have been measured as a function of associated charged-particle multiplicity for 3~Ar+ t97Au collisions at E/A = 35 MeV. Quantitative corrections for the dispersion of experimentally determined reaction planes about the true reaction plane have been applied. The results are compared to the predictions of a microscopic transport model which incorporates deuteron production and Coulomb effects.
During the early stages of a nuclear collision, the initially highly ordered m o t i o n o f projectile and target nucleons is dissipated and energy is fed into internal degrees o f freedom. The process d e p e n d s on the impact p a r a m e t e r o f the collision and on the transition rates within the reacting system. Evidence for the persistence o f collective m o t i o n throughout this nonequilibrium phase of the reaction [ 1-7] can be found in the enhanced emission o f energetic p a n i c l e s in the entrance-channel reaction plane [ 1,2,5-7]. A z i m u thal distributions predicted by the B o l t z m a n n - U c h i i n g - U h l e n b e c k ( B U U ) transport equation d e p e n d strongly on impact p a r a m e t e r [ 8,9 ], but such predictions have not yet been c o m p a r e d to multiplicity selected data. Q u a n t i t a t i v e c o m p a r i s o n s o f measured and calculated a z i m u t h a l distributions must account o f the resolution with which the reaction plane is determ i n e d experimentally. Standard techniques o f filtering theoretical predictions by the response o f the experimental apparatus are only valid for theories capable o f predicting the triple differential cross sect Present address: Instituto de Fisica, Universidade de Silo Paulo, C. Postal 20516, CEP 01498, Silo Paulo, Brazil. 2 Present address: Lawrence Berkeley Laboratory, Berkeley, CA 94720, USA. 3 Present address: Indiana Cyclotron Facility and Department of Chemistry, Indiana University, Bloomington, IN-47405, USA.
tions o f all particles e m p l o y e d in the reaction plane reconstruction. Since complex particles exhibit larger azimuthal anisotropies than nucleons [ 1,2,5-7 ], most experimental reaction plane reconstructions incorporate them into the analysis. This, however, makes it unfeasible to apply filtering techniques to microscopic transport models like the BUU theory in which cluster emission is not treated reliably. One can, however, extract the resolution o f the experimental reaction plane d e t e r m i n a t i o n from the d a t a and unfold the measured azimuthal distributions to obtain distributions with respect to the "true" reaction plane [7]. In this letter, we apply this new a p p r o a c h to extract the RMS angular widths of the proton and deuteron azimuthal distributions with respect to the true reaction plane as a function o f the associated charged particle multiplicity. By relating the charged particle multiplicity to the impact parameter o f the collision, we compare, for the first time, impact p a r a m e t e r (or multiplicity) selected experimental azimuthal distributions for protons and deuterons with microscopic calculations based upon the B U U theory [ 10,11 ]. The experiment was performed with a 35 M e V / nucleon 36Ar beam from the K1200 cyclotron o f Michigan State University b o m b a r d i n g a 197Au target o f i m g / c m 2 areal density. Charged p a n i c l e s were detected with the MSU Minibali [12], a plasticCsl (TI) phoswich detector array which covered an-
0370-2693/92/$ 05.00 © 1992 Elsevier Science Publishers B.V. All rights reserved.
243
Volume 297, number 3,4
PHYSICS LETTERS B
gles of 16 ° ~<0t~t,~ 160 °, corresponding to 85% of 4zt. Isotropic identification was achieved for p, d, t, SHe, and 4He. Heavier particles were identified by atomic number up to Z ~ 20. Representative energy thresholds for identified panicles are E,h/A ~ 2, 3, and 4 MeV for He, Ne, and Ar, respectively. Further experimental details are given in refs. [ 7,12,13 ]. Impact parameter averaged energy spectra for protons and deuterons detected at 0 = 4 5 ° , 72.5 ° , and 130 ° are shown in fig. I. While collective effects are difficult to unravel from the inclusive spectra [2], they can be readily discerned in the azimuthal emission patterns. To extract the azimuthal distributions of a given "trigger" panicle with respect to the "true" reaction plane, we have followed the procedure outlined in ref. [7]. For each event with N - l identified charged particles detected in coincidence with the trigger particle, we define [ 5-7 ] the "reconstructed" reaction plane as the plane spanned by the beam axis and the major axis of the transverse m o m e n t u m tensor for the N - 1 coincident particles ~'. The azimuthal angle, @, of the reconstructed reaction plane then minimizes the sum [ 5-7 ], '~ Excluding the "'trigger" particle avoids self-correlations. Corrections for momentum conservation effects were applied as discussed in ref. [7].
31 December 1992
N--I
Xp(q~)= Y. p2sin20~sin2(¢~,-@) .
(1)
/=1
In eq. ( 1 ), p,, 0, and q~, denote the momentum, polar and azimuthal angles o f particle i in the laboratory system. (The beam axis defines 0 = 0 °. ) As an example, the solid points in fig. 2 show "raw" azimuthal distributions, P,(O), relative to the reconstructed reaction plane [7] ~2. The left- and righthand panels show distributions measured for protons and deuterons, respectively, emitted at 0 = 4 5 ~ and selected by the multiplicity cut of N = 9. In order to provide a meaningful test o f microscopic transport models, one must minimize contributions of evaporative emission from equilibrated target-like residues. For this purpose, energy thresholds of E ( p ) >/32 McV and E(d) >t40 MeV were imposed on the trigger particle. Consistent with previous findings, emission is strongly enhanced in the reconstructed reaction plane; such collective effects are more pronounced for deuterons than for protons [ 1-7 ]. Due to fluctuations o f the angle between reconstructed and true reaction planes, the experimental azimuthal anisotropies, as shown in fig. 2, are smaller .2 For convenience, the distributions are normalized at P(0) = 1.
10 "
36Ar ¢ 197Au 1 ~ / A - 3 5 MeV '~,
• o-,o...~, e p
36A:._ l U ' A u 0 8 [o
:
•
_ 10_:3 ~_ .
x
.,
~.
•
"
7 o
"',
~< \ 45° T ' / - ~ ~. •. \ ,:% to /;-\, • \
10 -4
•
[3~
•
% "
P .
0
.
.
.
25
N=9
' "
50
,,I
. . . . . .
75
~,
0
d
\<30" ". \
~_
i
. . . . .
50
0 "45 = E(p) > 32 M e V
'(, i
'
25
0 ~I
75
I00
Fig. 1. Energy spectra for protons (left) and deuterons (right) detected at 0Utb=45 °, 72.5 ° and 130 °. The uncertainties in the data are smaller than the data points• The curves represent results from BUU calculations. They were renormalized by a multiplicative factor of 0.4 to make allowance for the emission of heavier clusters, not treated in the present calculations. (See footnote 4 ).
244
O4[!
, 130 °
o. .
_.
5 ~
'\
',, 10-5
2..,
..~.
'
72 5 °
,~,
~o 6
\~',45;!
~
\,
, ,~
o o L ~ ......... 0
45
[
90 0
E(d) > 40 M e V
45
90
Fig. 2. The points show azimuthal distributions, P,(¢), of protons (left) and deuterons (right) relative to the reconstructed reaction plane. The statistical uncertainties are smaller than the data points• The dot-dashed curves represent extracted distributions, P,(¢), relative to the true reaction plane. The solid curves depict the convolutions (eq. ( 2 ) ) of Pt(¢,) with P ( q 0 , the distributions of angles between the reconstructed and true reaction planes.
Volume 297, number 3,4
PHYSICS LETTERS B
than anisotropies with respect to the true reaction plane. Following refs. [ 7,1 4 ], we have estimated the dispersion o f reconstructed reaction planes about the true reaction plane by randomly subdividing each event into two sub-events, each containing half o f the N - 1 associated charged particles. By assuming statistical independence of the two reaction planes constructed from these two subevents, and analyzing the angular correlation between the two reaction planes, information about the distribution of the extracted reaction planes was obtained. Using this information, we approximated the correlation between, q~, the azimuthal angle o f the reaction plane extracted from the momenta o f the N - 1 associated particles, and, q)R, the azimuthal angle of the true reaction plane, by the expression, P( q ) - q~R) ~ Ci exp [ -- 002l sin 2 ( q)-- q)R ) ] , with 002 determined as a function of multiplicity N for each class of trigger particle. For the emitted light particles, the raw azimuthal distributions Pr(O) and the "true" distributions P,(O) with respect to the true reaction plane can be similarly related via the convolution [ 7 ]
31 December 1992
~,r., = <0 2 > ,/2 2
1/2
( f 0 P,(0) d0~ j
(3) '
and the integration is over the interval 0 ~ ~ ~ ½~. The results obtained for particles emitted at 0 = 4 5 ° and 72.5 ° are shown as solid points in fig. 3. Following ref. [ 13 ], we have assumed a monotonic relationship between charged particle multiplicity and impact parameter; the corresponding impact parameters are given as the upper scale. For both protons and deuterons, the extracted RMS widths increase with charged particle multiplicity, indicating increasingly isotropic distributions for decreasing impact parameters. Qualitatively this trend is expected. For impact parameter b = 0 , the emission must be azimuthally isotropic. However, the extracted RMS widths of the deuteron distributions at large N are smaller than q)rm~-~ 52 °, the RMS width of an isotropic distribution. This discrepancy may reflect some imprecision in the impact parameter selection. To explore whether the extracted widths can be reb (fro)
P~(0) = j" P,(O+O)P(O) dqO.
(2)
This relationship was obtained by assuming that the correlation between the angle of emission of the trigger particle and the experimentally extracted reaction plane is solely due to the correlation of each with the azimuthal angle, OR, of the true reaction plane. The experimental distributions can be accurately reproduced [7] by approximating P ( O ) by C~ × e x p ( - o 0 2 sin2q)), as indicated above, and P,(O) by C2 exp( -o02 sin2O), where o022 was determined by fitting the data with eq. (2). The resulting fits and extracted distributions P,(O) are given by the solid and clot-dashed curves in fig. 2, respectively. To allow a compact presentation of the multiplicity dependence of the azimuthal distributions, we have determined the RMS angular widths, qSrm~, of the true distributions P,(¢,), where ,3
u3 In the limit of very large values of N, the width would decrease by 1/,,/2, in accordance with the central limit theorem.
9.3 8.3
7 . 0 5 . 4 3.1
8.3
1.5
7.05.4
3,1 1.5
45 40 35
~
30
i
E(p) > 32 MeV i
E(d) > 4 0 M e V
i
45 40 Q Onn = Ofree 0 0 n n = .5Ofree
35 30
5
10
i 15
5
1LO
15
N
Fig. 3. Solid points: Multiplicity dependence of true azimuthal widths of protons (left-hand panels) and deuterons (right-hand panels) emitted at 0=45 ° (top panels) and 0=72.5 ° (bottom panels). The experimental uncertainties, largely systematic, have been estimated by a Monte Carlo simulation of the experiment [ 15 ]. Open points: impact parameter dependence predicted by BUU calculations using a stiffequation of state and a,, =0.5afr~ (diamonds) and a,,,=au~ (squares). The multiplicity and impact parameter scales in the figure are related according to the geometrical prescription of ref. [ 131. 245
Volume 297, number 3,4
PHYSICS LETTERS B
produced by microscopic transport models, we performed calculations within the framework of a modified BUU model [ i 1 ] capable of predicting proton and deuteron production. This model provides reasonable descriptions ofd/p ratios and of energy spectra at bombarding energies of 100 MeV£E/A<,2 GeV [ l I ]. As in other work [ 8-10 ], we adopted a parametrization of the optical potential in terms o f a local density, representing a stiff equation of state (EOS) with compressibility K = 380 MeV. Coulomb interactions were taken into account and the in-medium cross section ann was taken to be proportional to trf,~, the experimentally determined free nucleonnucleon cross section. The calculations were terminated at the collision time of 220 fm/c. The emitted protons and deuterons were subsequently Coulomb accelerated according to the local Coulomb field which was assumed to be time independent in the center-of-momentum system. For particles with energies near the semi-classical Coulomb barrier, this approximation provides Coulomb trajectories which differ slightly from the exact trajectories. The approximation is, however, reasonably accurate for protons and deuterons with energies above the lowenergy cuts used in the angular correlations shown in figs. 2 and 3. Inclusive spectra predicted by the model are shown in fig. 1. The predicted differential cross sections were adjusted to the data by multiplication with a factor o f 0.4. This renormalization is reasonable because the present calculations do not include the loss o f flux due to emission o f heavier clusters (t, 3He, a, ...) ~4. The calculated relative cross sections are in fair agreement with the data, At 0 = 45 °, the calculations predict somewhat steeper spectra than observed experimentally; at 0 = 130 °, the trend is reversed. At this latter angle, the BUU model significantly overprediets the preequilibrium nucleon flux, a trend which has been observed in other simulations of intermediate energy collisions [8,16], but is not yet understood. Fig. 3, depicts the impact parameter dependence of the predicted widths. Open squares and diamonds show the results of calculations using trn~= efr~ and a..=½trrr~, respectively. The predicted azimuthal For 0,.b=45 °, the total proton flux, including protons bound in heavier fragments, is approximately 2.5 times the sum of the proton and deuteron fluxes. 246
31 December 1992
RMS angular widths are not subject to normalization uncertainties; they are comparable to those experimentally observed. Like the experimental data, the calculations predict smaller ~ , for deuterons than for protons, and smaller ~,~s at backward angles than for forward angles. Nevertheless, the calculations quantitatively disagree with the data. In particular, for b > 4 fm they predict a more gradual dependence of ~ s upon b than is observed. While some uncertainty exists with regard to the relation between the multiplicity and the impact parameter [ 13 ], impact parameter averaging the calculations will not change the trend at b > 4 fm where the calculations depend less strongly upon the impact parameter than do the data. Since the differences between the two sets of calculations are comparable to the differences between experiment and theory, different choices of the transport model parameters or a different treatment of the mean field in the nuclear surface may be required to reconcile these discrepancies. Due to the large amounts o f computer time required, however, further investigations of model sensitivities lie beyond the scope of the present paper. In summary, proton and deuteron azimuthal distributions for 36Ar induced reactions on t97Au have been measured relative to the entrance-channel reaction plane. These distributions are strongly enhanced in the reaction plane for peripheral collisions, but become more isotropic for central collisions. The present calculations reproduce the overall magnitude of the observed azimuthal anisotropies. However, discrepancies between theory and experiment are sufficiently large to warrant further theoretical work. This paper is based upon work supported by the National Science Foundation under Grant numbers PHY-84-51280, PHY-89-13815, PHY-89-05933 and PHY-90-17077. W.G.L. acknowledges the receipt of a US Presidential Young Investigator Award.
References
31 ] M.B. Tsang et al., Phys. Rev. Left. 52 (1984) 1967. [2] C.B. Chitwood et al., Phys. Rev. C 34 (1986) 858. [3] M.B. Tsang et al.. Phys. Rev. Lett. 57 (1986) 559. [4] M.B. Tsang et al., Phys. Rev. Lett. 60 (1988) 1479. [ 5 ] W.K. Wilson et al., Phys. Rev. C 41 (1990) R 1881. [6] W.K. Wilson et al., Phys. Rev. C45 (1992) 738. 17] M.B. Tsang et al., Phys. Rev. C 44 ( 1991 ) 2065.
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PHYSICS LETTERS B
[ 8 ] M.B. Tsang et al., Phys. Rev. C 40 (1989) 1685. [9] H.M. Xu et al, Phys. Rev. Lett. 65 (1990) 843. [ 10] G.F. Bertsch and S. Das Gupta, Phys. Rep. 160 (1988) 189, and references therein. [ 11 ] P. Danielewicz and G.F. Berlsch, Nucl. Phys. A 533 ( 1991 ) 712. [ 12] R.T. de Souza et al., Nucl. lnstrum. Methods A 295 (1990) 109.
31 December 1992
[ 13 ] Y.D. Kim et al, Phys. Rev. C 45 (1992) 338. [ 14] P. Danielewicz and G. Odyniec, Phys. Lett. B 157 (1985) 146. [ 15 ] M.B. Tsang et al., to be published. [ 16] J. Aichelin, Phys. Rev. C 33 (1986) 537; W.G. Gong et al., Phys. Rev. C43 (1991) 1804.
247
PHYSICS LETTERS B
Reaction dynamics and deuteron production M.B. Tsang, P. Danielewicz, D.R. Bowman, N. Carlin ~, C.K. Gelbke, W.G. Gong 2, Y.D. Kim 3, W.G. Lynch, L. Phair, R.T. de Souza 3 and F. Zhu National Superconducting Cyclotron Laboratory and Department of Physics and Astronomy, Michtgan State University, East Lansing, M148824, USA Received 23 April 1992; revised manuscript received 16 October 1992
Proton and deuteron azimuthal distributions with respect to the reaction plane have been measured as a function of associated charged-particle multiplicity for 3~Ar+ t97Au collisions at E/A = 35 MeV. Quantitative corrections for the dispersion of experimentally determined reaction planes about the true reaction plane have been applied. The results are compared to the predictions of a microscopic transport model which incorporates deuteron production and Coulomb effects.
During the early stages of a nuclear collision, the initially highly ordered m o t i o n o f projectile and target nucleons is dissipated and energy is fed into internal degrees o f freedom. The process d e p e n d s on the impact p a r a m e t e r o f the collision and on the transition rates within the reacting system. Evidence for the persistence o f collective m o t i o n throughout this nonequilibrium phase of the reaction [ 1-7] can be found in the enhanced emission o f energetic p a n i c l e s in the entrance-channel reaction plane [ 1,2,5-7]. A z i m u thal distributions predicted by the B o l t z m a n n - U c h i i n g - U h l e n b e c k ( B U U ) transport equation d e p e n d strongly on impact p a r a m e t e r [ 8,9 ], but such predictions have not yet been c o m p a r e d to multiplicity selected data. Q u a n t i t a t i v e c o m p a r i s o n s o f measured and calculated a z i m u t h a l distributions must account o f the resolution with which the reaction plane is determ i n e d experimentally. Standard techniques o f filtering theoretical predictions by the response o f the experimental apparatus are only valid for theories capable o f predicting the triple differential cross sect Present address: Instituto de Fisica, Universidade de Silo Paulo, C. Postal 20516, CEP 01498, Silo Paulo, Brazil. 2 Present address: Lawrence Berkeley Laboratory, Berkeley, CA 94720, USA. 3 Present address: Indiana Cyclotron Facility and Department of Chemistry, Indiana University, Bloomington, IN-47405, USA.
tions o f all particles e m p l o y e d in the reaction plane reconstruction. Since complex particles exhibit larger azimuthal anisotropies than nucleons [ 1,2,5-7 ], most experimental reaction plane reconstructions incorporate them into the analysis. This, however, makes it unfeasible to apply filtering techniques to microscopic transport models like the BUU theory in which cluster emission is not treated reliably. One can, however, extract the resolution o f the experimental reaction plane d e t e r m i n a t i o n from the d a t a and unfold the measured azimuthal distributions to obtain distributions with respect to the "true" reaction plane [7]. In this letter, we apply this new a p p r o a c h to extract the RMS angular widths of the proton and deuteron azimuthal distributions with respect to the true reaction plane as a function o f the associated charged particle multiplicity. By relating the charged particle multiplicity to the impact parameter o f the collision, we compare, for the first time, impact p a r a m e t e r (or multiplicity) selected experimental azimuthal distributions for protons and deuterons with microscopic calculations based upon the B U U theory [ 10,11 ]. The experiment was performed with a 35 M e V / nucleon 36Ar beam from the K1200 cyclotron o f Michigan State University b o m b a r d i n g a 197Au target o f i m g / c m 2 areal density. Charged p a n i c l e s were detected with the MSU Minibali [12], a plasticCsl (TI) phoswich detector array which covered an-
0370-2693/92/$ 05.00 © 1992 Elsevier Science Publishers B.V. All rights reserved.
243
Volume 297, number 3,4
PHYSICS LETTERS B
gles of 16 ° ~<0t~t,~ 160 °, corresponding to 85% of 4zt. Isotropic identification was achieved for p, d, t, SHe, and 4He. Heavier particles were identified by atomic number up to Z ~ 20. Representative energy thresholds for identified panicles are E,h/A ~ 2, 3, and 4 MeV for He, Ne, and Ar, respectively. Further experimental details are given in refs. [ 7,12,13 ]. Impact parameter averaged energy spectra for protons and deuterons detected at 0 = 4 5 ° , 72.5 ° , and 130 ° are shown in fig. I. While collective effects are difficult to unravel from the inclusive spectra [2], they can be readily discerned in the azimuthal emission patterns. To extract the azimuthal distributions of a given "trigger" panicle with respect to the "true" reaction plane, we have followed the procedure outlined in ref. [7]. For each event with N - l identified charged particles detected in coincidence with the trigger particle, we define [ 5-7 ] the "reconstructed" reaction plane as the plane spanned by the beam axis and the major axis of the transverse m o m e n t u m tensor for the N - 1 coincident particles ~'. The azimuthal angle, @, of the reconstructed reaction plane then minimizes the sum [ 5-7 ], '~ Excluding the "'trigger" particle avoids self-correlations. Corrections for momentum conservation effects were applied as discussed in ref. [7].
31 December 1992
N--I
Xp(q~)= Y. p2sin20~sin2(¢~,-@) .
(1)
/=1
In eq. ( 1 ), p,, 0, and q~, denote the momentum, polar and azimuthal angles o f particle i in the laboratory system. (The beam axis defines 0 = 0 °. ) As an example, the solid points in fig. 2 show "raw" azimuthal distributions, P,(O), relative to the reconstructed reaction plane [7] ~2. The left- and righthand panels show distributions measured for protons and deuterons, respectively, emitted at 0 = 4 5 ~ and selected by the multiplicity cut of N = 9. In order to provide a meaningful test o f microscopic transport models, one must minimize contributions of evaporative emission from equilibrated target-like residues. For this purpose, energy thresholds of E ( p ) >/32 McV and E(d) >t40 MeV were imposed on the trigger particle. Consistent with previous findings, emission is strongly enhanced in the reconstructed reaction plane; such collective effects are more pronounced for deuterons than for protons [ 1-7 ]. Due to fluctuations o f the angle between reconstructed and true reaction planes, the experimental azimuthal anisotropies, as shown in fig. 2, are smaller .2 For convenience, the distributions are normalized at P(0) = 1.
10 "
36Ar ¢ 197Au 1 ~ / A - 3 5 MeV '~,
• o-,o...~, e p
36A:._ l U ' A u 0 8 [o
:
•
_ 10_:3 ~_ .
x
.,
~.
•
"
7 o
"',
~< \ 45° T ' / - ~ ~. •. \ ,:% to /;-\, • \
10 -4
•
[3~
•
% "
P .
0
.
.
.
25
N=9
' "
50
,,I
. . . . . .
75
~,
0
d
\<30" ". \
~_
i
. . . . .
50
0 "45 = E(p) > 32 M e V
'(, i
'
25
0 ~I
75
I00
Fig. 1. Energy spectra for protons (left) and deuterons (right) detected at 0Utb=45 °, 72.5 ° and 130 °. The uncertainties in the data are smaller than the data points• The curves represent results from BUU calculations. They were renormalized by a multiplicative factor of 0.4 to make allowance for the emission of heavier clusters, not treated in the present calculations. (See footnote 4 ).
244
O4[!
, 130 °
o. .
_.
5 ~
'\
',, 10-5
2..,
..~.
'
72 5 °
,~,
~o 6
\~',45;!
~
\,
, ,~
o o L ~ ......... 0
45
[
90 0
E(d) > 40 M e V
45
90
Fig. 2. The points show azimuthal distributions, P,(¢), of protons (left) and deuterons (right) relative to the reconstructed reaction plane. The statistical uncertainties are smaller than the data points• The dot-dashed curves represent extracted distributions, P,(¢), relative to the true reaction plane. The solid curves depict the convolutions (eq. ( 2 ) ) of Pt(¢,) with P ( q 0 , the distributions of angles between the reconstructed and true reaction planes.
Volume 297, number 3,4
PHYSICS LETTERS B
than anisotropies with respect to the true reaction plane. Following refs. [ 7,1 4 ], we have estimated the dispersion o f reconstructed reaction planes about the true reaction plane by randomly subdividing each event into two sub-events, each containing half o f the N - 1 associated charged particles. By assuming statistical independence of the two reaction planes constructed from these two subevents, and analyzing the angular correlation between the two reaction planes, information about the distribution of the extracted reaction planes was obtained. Using this information, we approximated the correlation between, q~, the azimuthal angle o f the reaction plane extracted from the momenta o f the N - 1 associated particles, and, q)R, the azimuthal angle of the true reaction plane, by the expression, P( q ) - q~R) ~ Ci exp [ -- 002l sin 2 ( q)-- q)R ) ] , with 002 determined as a function of multiplicity N for each class of trigger particle. For the emitted light particles, the raw azimuthal distributions Pr(O) and the "true" distributions P,(O) with respect to the true reaction plane can be similarly related via the convolution [ 7 ]
31 December 1992
~,r., = <0 2 > ,/2 2
1/2
( f 0 P,(0) d0~ j
(3) '
and the integration is over the interval 0 ~ ~ ~ ½~. The results obtained for particles emitted at 0 = 4 5 ° and 72.5 ° are shown as solid points in fig. 3. Following ref. [ 13 ], we have assumed a monotonic relationship between charged particle multiplicity and impact parameter; the corresponding impact parameters are given as the upper scale. For both protons and deuterons, the extracted RMS widths increase with charged particle multiplicity, indicating increasingly isotropic distributions for decreasing impact parameters. Qualitatively this trend is expected. For impact parameter b = 0 , the emission must be azimuthally isotropic. However, the extracted RMS widths of the deuteron distributions at large N are smaller than q)rm~-~ 52 °, the RMS width of an isotropic distribution. This discrepancy may reflect some imprecision in the impact parameter selection. To explore whether the extracted widths can be reb (fro)
P~(0) = j" P,(O+O)P(O) dqO.
(2)
This relationship was obtained by assuming that the correlation between the angle of emission of the trigger particle and the experimentally extracted reaction plane is solely due to the correlation of each with the azimuthal angle, OR, of the true reaction plane. The experimental distributions can be accurately reproduced [7] by approximating P ( O ) by C~ × e x p ( - o 0 2 sin2q)), as indicated above, and P,(O) by C2 exp( -o02 sin2O), where o022 was determined by fitting the data with eq. (2). The resulting fits and extracted distributions P,(O) are given by the solid and clot-dashed curves in fig. 2, respectively. To allow a compact presentation of the multiplicity dependence of the azimuthal distributions, we have determined the RMS angular widths, qSrm~, of the true distributions P,(¢,), where ,3
u3 In the limit of very large values of N, the width would decrease by 1/,,/2, in accordance with the central limit theorem.
9.3 8.3
7 . 0 5 . 4 3.1
8.3
1.5
7.05.4
3,1 1.5
45 40 35
~
30
i
E(p) > 32 MeV i
E(d) > 4 0 M e V
i
45 40 Q Onn = Ofree 0 0 n n = .5Ofree
35 30
5
10
i 15
5
1LO
15
N
Fig. 3. Solid points: Multiplicity dependence of true azimuthal widths of protons (left-hand panels) and deuterons (right-hand panels) emitted at 0=45 ° (top panels) and 0=72.5 ° (bottom panels). The experimental uncertainties, largely systematic, have been estimated by a Monte Carlo simulation of the experiment [ 15 ]. Open points: impact parameter dependence predicted by BUU calculations using a stiffequation of state and a,, =0.5afr~ (diamonds) and a,,,=au~ (squares). The multiplicity and impact parameter scales in the figure are related according to the geometrical prescription of ref. [ 131. 245
Volume 297, number 3,4
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produced by microscopic transport models, we performed calculations within the framework of a modified BUU model [ i 1 ] capable of predicting proton and deuteron production. This model provides reasonable descriptions ofd/p ratios and of energy spectra at bombarding energies of 100 MeV£E/A<,2 GeV [ l I ]. As in other work [ 8-10 ], we adopted a parametrization of the optical potential in terms o f a local density, representing a stiff equation of state (EOS) with compressibility K = 380 MeV. Coulomb interactions were taken into account and the in-medium cross section ann was taken to be proportional to trf,~, the experimentally determined free nucleonnucleon cross section. The calculations were terminated at the collision time of 220 fm/c. The emitted protons and deuterons were subsequently Coulomb accelerated according to the local Coulomb field which was assumed to be time independent in the center-of-momentum system. For particles with energies near the semi-classical Coulomb barrier, this approximation provides Coulomb trajectories which differ slightly from the exact trajectories. The approximation is, however, reasonably accurate for protons and deuterons with energies above the lowenergy cuts used in the angular correlations shown in figs. 2 and 3. Inclusive spectra predicted by the model are shown in fig. 1. The predicted differential cross sections were adjusted to the data by multiplication with a factor o f 0.4. This renormalization is reasonable because the present calculations do not include the loss o f flux due to emission o f heavier clusters (t, 3He, a, ...) ~4. The calculated relative cross sections are in fair agreement with the data, At 0 = 45 °, the calculations predict somewhat steeper spectra than observed experimentally; at 0 = 130 °, the trend is reversed. At this latter angle, the BUU model significantly overprediets the preequilibrium nucleon flux, a trend which has been observed in other simulations of intermediate energy collisions [8,16], but is not yet understood. Fig. 3, depicts the impact parameter dependence of the predicted widths. Open squares and diamonds show the results of calculations using trn~= efr~ and a..=½trrr~, respectively. The predicted azimuthal For 0,.b=45 °, the total proton flux, including protons bound in heavier fragments, is approximately 2.5 times the sum of the proton and deuteron fluxes. 246
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RMS angular widths are not subject to normalization uncertainties; they are comparable to those experimentally observed. Like the experimental data, the calculations predict smaller ~ , for deuterons than for protons, and smaller ~,~s at backward angles than for forward angles. Nevertheless, the calculations quantitatively disagree with the data. In particular, for b > 4 fm they predict a more gradual dependence of ~ s upon b than is observed. While some uncertainty exists with regard to the relation between the multiplicity and the impact parameter [ 13 ], impact parameter averaging the calculations will not change the trend at b > 4 fm where the calculations depend less strongly upon the impact parameter than do the data. Since the differences between the two sets of calculations are comparable to the differences between experiment and theory, different choices of the transport model parameters or a different treatment of the mean field in the nuclear surface may be required to reconcile these discrepancies. Due to the large amounts o f computer time required, however, further investigations of model sensitivities lie beyond the scope of the present paper. In summary, proton and deuteron azimuthal distributions for 36Ar induced reactions on t97Au have been measured relative to the entrance-channel reaction plane. These distributions are strongly enhanced in the reaction plane for peripheral collisions, but become more isotropic for central collisions. The present calculations reproduce the overall magnitude of the observed azimuthal anisotropies. However, discrepancies between theory and experiment are sufficiently large to warrant further theoretical work. This paper is based upon work supported by the National Science Foundation under Grant numbers PHY-84-51280, PHY-89-13815, PHY-89-05933 and PHY-90-17077. W.G.L. acknowledges the receipt of a US Presidential Young Investigator Award.
References
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PHYSICS LETTERS B
[ 8 ] M.B. Tsang et al., Phys. Rev. C 40 (1989) 1685. [9] H.M. Xu et al, Phys. Rev. Lett. 65 (1990) 843. [ 10] G.F. Bertsch and S. Das Gupta, Phys. Rep. 160 (1988) 189, and references therein. [ 11 ] P. Danielewicz and G.F. Berlsch, Nucl. Phys. A 533 ( 1991 ) 712. [ 12] R.T. de Souza et al., Nucl. lnstrum. Methods A 295 (1990) 109.
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[ 13 ] Y.D. Kim et al, Phys. Rev. C 45 (1992) 338. [ 14] P. Danielewicz and G. Odyniec, Phys. Lett. B 157 (1985) 146. [ 15 ] M.B. Tsang et al., to be published. [ 16] J. Aichelin, Phys. Rev. C 33 (1986) 537; W.G. Gong et al., Phys. Rev. C43 (1991) 1804.
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