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Coalescence-model analysis of alpha-particle and deutron spectra from energetic heavy-ion collisions
Volume 192, number 3,4
PHYSICS LETTERS B
2 July 1987
COALESCENCE-MODEL ANALYSIS OF ALPHA-PARTICLE AND DEUTERON SPECTRA FROM ENERGETIC HEAVY-ION COLLISIONS S. D A T T A , R. (~APLAR, N. C I N D R O Rudjer Bo2kovi6Institute, 41001 Zagreb, Croatia, Yugoslavia R.L. AUBLE, J.B. BALL A N D R.L. R O B I N S O N Oak Ridge National Laboratory, Oak Ridge, TN 37831, USA Received 2 January 1987; revised manuscript received 10 April 1987
Double differential cross sections for alpha particles and deuterons emitted from various colliding nuclear systems are analysed in terms of the coalescence model. The systems vary from light, symmetric to heavy, asymmetric ones; projectile energies are 15-25 MeV/nucleon. The obtained values of coalescence radii in momentum space show a striking regularity with A/(Aj,+AT).
The coalescence m o d e l o f complex-particle production in nuclear reactions has been successfully used at high b o m b a r d i n g energies. The m o d e l as form u l a t e d in refs. [ 1,2] accounts for the f o r m a t i o n o f high-energy deuterons from targets exposed to 2 5 - 3 0 GeV protons. The p r o p o s e d f o r m a t i o n m e c h a n i s m considers the pairing o f nucleons from cascades which develop in the struck nucleus. Basic to the idea is the concept o f coalescence radius [2] as the radius o f a small sphere a r o u n d the p o i n t representing any nucleon in m o m e n t u m space. Any n u m b e r o f nucleons whose relative m o m e n t u m is smaller than the coalescence radius forms a complex particle which is then emitted. Recently, several authors [ 3 - 5 ] have used the coalescence idea to explain the spectra o f complex particles, d, t, 3He, and ct's, e m i t t e d in heavy-ion reactions at energies a r o u n d 20 MeV/nucleon. This application, however, is not without conceptual and practical difficulties. In the present p a p e r we refine the coalescence model so as to make it m o r e a p p r o p r i a t e for use at energies o f 15-25 MeV/nucleon. In this way we obtain information on the systematic b e h a v i o r o f coalescence radii with system masses at these energies. We start by assuming that p r e - e q u i l i b r i u m emis302
sion o f nucleons in reactions induced by nucleons o f as low an energy as 15-25 MeV can be related to cascades initiated by these nucleons in the target nucleus. Thus, the coalescence-model ideas imply that the coalescence takes place among nucleons participating in the first few collisions, i.e., before the equilibrium sets in. In our t r e a t m e n t we correctly take into account the effects o f the C o u l o m b barrier by calculating the C o u l o m b repulsion between the e m i t t e d particles a n d the emitting system. Such a correction was i n t r o d u c e d by Awes et al. [3,4] as a fitting parameter. F u r t h e r m o r e , we properly treat the influence o f the nuclear-matter e n v i r o n m e n t on the formation o f the complex particle by introducing a mass correction which takes into account its b i n d i n g energy. The analysed d a t a consist o f double differential cross sections o f protons, deuterons, and ct particles e m i t t e d from the colliding systems listed in table 1, m e a s u r e d at l a b o r a t o r y angles o f 10 °, 30 °, 50 °, and 70 ° [6] #1 The derivation o f the model formulas used for data analysis follows the lines o f refs. [3,4]; we have, however, carried it out by including the complexparticle b i n d i n g energy; hence, now ma ¢:A.mp. The "~ Data were obtained at the Holifield Heavy Ion Research Facility, ORNL, Oak Ridge, TN.
0370-2693/87/$ 03.50 © Elsevier Science Publishers B.V. ( N o r t h - H o l l a n d Physics Publishing D i v i s i o n )
Volume
192, number
3,4
Table 1 Table of system- and angle-average System
PHYSICS
coalescence
LETTERS
B
2 July 1987
radii. Average coalescence
&,IA (MeV)
radius ( MeVlc)
angle (degrees)
system
10
30
50
70
alphas
j60+ “Al ‘hO+‘hTi ‘hO+6”Ni Ib0+ “OSn 160+ 19’Au “S + ‘,A, 3’S+ ?,A, 3ZS+ 4~Ti 32s + 46Ti “S + h”Ni 3’S+ “7AU 3’S+ “7AU S”Ni + h”Ni 58Ni + I‘“Sn SnNi+ “jSn 5”Ni + “‘Au
25.2 25.2 25.2 25.5 25.2 15.7 21.2 15.7 21.2 15.7 15.7 21.2 15.1 15.1 15.1 15.1
80 80 80 91 109 67 90 65 67 69 90 103 65 78 80 99
84 86 85 95 100 66 94 71 71 72 94 90 68 82 68 98
99 105 103 118 122 86 90 85 93 90 107 110 86 104 104 116
90 101 96 123 129 85 91 88 89 87 116 109 87 105 100 113
86 92 88 101 113 74 75 78 77 78 99 101 73 90 85 104
deuterons
“0 + 4hTi IhO+ 19’Au “S+ ‘07AU 58Ni + “ONi
25.2 25.2 21.2 15.1
64 90 87 46
68 87 81 48
87 112 103 71
81 125 105 69
73 101 92 57
cross section for the emission of complex particles of mass number A=N+ 2 can be obtained from the proton-emission spectra using the relation [ 7-101
with VZ.,~ the mass of the complex particle andf, Coulomb repulsion energy,
($),,=A (2)”
x4= 1.29(AT+Ap
(1) where N, Np, and NT. are, respectively, the neutron numbers of the complex particle, the projectile, and the target nucleus, and, similarly, 2, Zp, and 2, are the corresponding proton numbers. PO is the coalescence radius discussed earlier and otO, is the total cross section, for which we use the expression from ref. [ 111. As a next step, we assume that the complex particle is formed near the surface of the emitting nucleus. Then, denoting the momentum of the complex particle at the nuclear surface and when detected by P,4,, and PA, respectively, we have P.:/2m,d = P&J2m,4 +fA ,
(2)
1.44(Z,$Z,-Z)Z
the
MeV -A)“3
’
(3)
We also assume that the direction of the emitted particle is not affected by the Coulomb field. Then it can be shown that double differential cross sections for the emission of complex particles (d’ald&dQ), can be calculated from the measured proton cross sections (d’aldE,dS2),,,,,, by using the expression
with
Volume 192, number 3,4
PHYSICS LETTERSB
where p is the proton m o m e n t u m in the laboratory system, mp its mass, and fp the Coulomb barrier for protons [eq. (3) with A = Z = I ] . The relation between the laboratory energy AA of the complex particle A and that of the proton, Ep, is given by
2 July 1987
10"s
)0 "~
~ llD~i~llh'~ 0(3 "Vil~
x.~
E.4 = Azmp (Ep - f p ) +fro • mA
(6)
The importance of taking into account the effects of the Coulomb barrier and that of the particle binding energy, as done explicitly in eq. (6), can be readily recognized by considering the ratios EA/Ep of energies at which the observed complex-particle and proton spectra should be compared. This ratio influences expression (4) via the corresponding double differential cross sections which fall off exponentially with the energy of the emitted particle. For alpha particles, for instance, the ratio changes from E A / E p = 4 at very high energies (and taking mA = 4mp) to EA/Ep= 3-3.5 at spectral energies encountered in the present analysis and taking into account the correct mass m~= 4 m p - 2 6 . 7 MeV/c 2. This refinement considerably modifies the value of C given by (4), hence the values of Po extracted from (5). The analysis of the experimental data proceeds as follows. We first check the physical consistency of the values of the coalescence radius obtained for each system at various angles and energies. This we do by taking the ratio C as defined in expression (4), with the double differential cross sections representing the measured alpha (A = 4) and proton energy spectra, respectively; similarly, for deuterons. Then, using eq. (5) we extract the corresponding values of the coalescence radius Po in the following way. For a colliding system, we obtain the values of C from the spectra at various angles at energy intervals AEp = 1-2 MeV and the corresponding AE~=AE~ and AEA=AEd [see eq. (6)]. Thus a series of values of Po are obtained for each colliding system and each angle. The observation we make is that for the higherenergy part of the spectra, above some minimal energy, say, E min, the values obtained for P0 lie within _+5% of the average value for a given angle. At the most forward angle (10 ° ) this minimal energy lies well above 1.5-2 times the energy of beam-velocity alphas and deuterons. At 70 ° E ~ is 3-4 times above the corresponding values of the Coulomb barrier fA. In this way we obtain what we call the angle-average
coalescence radii. 304
SYSTEM
1
3ZS+ '97Au--'-X + d E,n{ = 6?9.8
li
O SYSIEM AVERAGE R DIUS X ANGLE AVERAGERADIUS • EXPERIMENT
-,,~
xx b ~
10.3
10.2
10,~
i
10 o
10-~
10-~
10-3
J 10
30
i
,
SO
10
?
;
~
90
I
h
110
~
. . . . . .
130
150
170
190
Ed (MeV)
Fig. 1. Deuteron spectra from 32S+ 197Auat E~nc=679.8 MeV. Finally, we take the statistically weighted average of all angle-average coalescence radii and obtain the coalescence radius for a given system at a given incident energy which we call the system-average coalescence radius. These radii are shown in the last column of table 1. Fig. 1 shows the experimental deuteron spectra for the 328i-k- 197Au system at 679.8 MeV and the spectra calculated by using (i) angle-average coalescence radii obtained, respectively, at 0~ab= 10 °, 30 °, 50 °, and 70 °, and (ii) the system-average radius. There is overall agreement with the experimental data, as well as quite satisfactory agreement between the two sets of calculated values, which demonstrates the consistency of our procedure. The agreement with
Volume 192, number 3,4
PHYSICS LETTERS B
140
120
~x~ ALPHAS
C~
Ap [SMALL AP+AT L LARGE
,-g
>~ lOO
~ %o -..~ ~'°-. ~ o "--a..
o') ¢0 e~ 0
tJ m
120" ~. %
0
~ 100
x%,
0 SYSTEM AVERAGE RADIUS
DEUTERONS
AVERAGES FOR: o ANGLES 10" AND 30' z~ ANGLES 50' AND 70'
80
60
40
0.1
0.2
0.3
0.4
0.5
0.6
0.7
Ap/(Ap+AO Fig. 2. Extracted coalescence radii Po versus Ap/(Ap+AT) for various colliding systems. the data is better for the high-energy part of the spectra, as expected. Similar results are obtained for deuteron and a spectra for all other studied systems. The values of the coalescence radii for alpha particles and deuterons obtained in the analysis are plotted versus Ap/(Ap +AT) in fig. 2. Two features of this figure deserve our attention. Firstly, the system-average values obtained for coalescence radii display a striking regularity with Ap/(Ap+AT): the coalescence radii increase with decreasing values of this parameter (larger Po for asymmetric colliding systems). The effect, seen for both alphas and deuterons, can be approximated by a parabolic best-fit curve (full lines). Secondly, as mentioned earlier, the angle-average coalescence radii obtained at more forward angles (10 ° and 30 °) are systematically smaller than those obtained at larger angles (50 ° and 70 ° ) for the same system. To point out this feature, in fig. 2 we have plotted separately arithmetic means of 10 ° and 30 ° and of 50 ° and 70 °
2 July 1987
radii for each analysed system. They maintain the parabolic trend of the system radii, but with two distinct parabolas (dashed lines). The decrease of Po with increasing Ap/(Ap+AT), i.e., when moving to more symmetric colliding systems, is expected from a crude picture of the process illustrated by the inset in fig. 2. In collisions of systems with large asymmetry [small values of Ar,/(Ap+AT)], the fraction of particles emitted in central collisions is expected to be larger than for symmetric systems. In the former, hence, a cascade nucleon finds a sufficient number of nucleons moving in its vicinity within a given direction to form a complex particle to be emitted. In symmetric systems, however, a cascade nucleon in the periphery will find a lesser number of nucleons suitable for coalescing and forming a complex particle, hence will require a larger volume radius (smaller momentum radius) to do so. Thus an effectively smaller coalescence radius Po for symmetric Systems. It is worth noting that in the studied Ei,c/A range (15-25 MeV/nucleon) system coalescence radii obtained from the model display little, if any, energy dependence. The deuteron radii P0 obtained from the analysis compare well with values reported in the literature; the alpha radii, however, are somewhat smaller [ 12 ]. A thermodynamic calculation based on ref. [10] gives the radius R of a thermalized volume corresponding to the momentum-space coalescence radius P0 as
R = { [Z!N~.A3(2s+1 )exp(Eo/T)]/2 A } ~/3(A ~) × (3/4n) 2/3h/Po ,
(7)
where Eo is the binding energy of the composite particle and s its spin. Using T = 11 MeV (ref. [ 13]) and a mean v a l u e / o = 9 0 MeV/c, we obtain R~---9.5 fm for alpha particles. Similarly, with Po = 80 MeV/c, we obtain Rd= 11.6 fm for deuterons. The latter value compares well with that reported in ref. [ 14]. The meaningfulness of such large spatial radii, encompassing essentially the whole nucleus, should be questioned and, with it, the applicability of eq. (7) to the energy range E / A < 100 MeV/u. The uncertainty principle defines the spatial coalescencesphere radius as
305
Volume 192, number 3,4
R <~h/Po ,
PHYSICS LETTERS B (8)
yielding ~ 2 . 2 fm for alphas a n d ~ 2 . 5 fm for deuterons. Such radii are more consistent with what we know about the n u c l e o n - n u c l e o n interaction. Concluding, we should say that (i) we have refined the coalescence model by taking into account the b i n d i n g energy in the masses of the complex ejectiles and applied it to the analysis of alpha a n d deuteron spectra from heavy-ion collisions. This correction a n d the properly handled Coulomb-barrier correction have a considerable influence on the extracted values of the coalescence radii Po; (ii) the extracted values of Po show a striking dependence on Ap/(Ap+AT), increasing regularly from symmetric to asymmetric colliding systems. This dependence is compatible with a geometrical picture of the reaction mechanism. This paper is part of a study u n d e r the US-Yugoslav collaboration project JFP 554 DOE/IRB. The financial support of the j o i n t
306
2 July 1987
US-Yugoslav Board for Scientific a n d Technological Cooperation is gratefully acknowledged.
References [1] S.T. Butler and C.A. Pearson, Phys. Rev. 129 (1963) 836. [2] A. Schwarzschild and ~. Zupan~i6, Phys. Rev. 129 (1963) 854. [3] T.C. Awes et al., Phys. Rev. C 24 (1981) 89. [4] T.C. Aweset al., Phys. Rev. C 25 (1982) 2361. [51 H. Machner et al., Phys. Rev. C 31 (1985) 443. [6] R.L. Auble et al., unpublished. [7] H.H. Gutbrod et al., Phys. Rev. Lett. 37 (1976) 667. [8] J. Gosset et al., Phys. Rev. C 16 (1977) 629. [9] M.C. Lemaire et al., Phys. Lett. B 85 (1979) 38. [10] A.Z. Mekjian, Phys. Rev. C 17 (1978) 1051. [ 11 ] S. Nagamiya et al., Phys. Rev. C 24 (1981) 971. [ 12] A. Fahli et al., Z. Phys. A 326 (1987) 169. [13] M. Korolija et al., Proc. XVI Intern. Symp. on Nuclear physics (Gaussig, GDR, 1986), eds. R. Reif et al., to be published. [ 14] R.L. Auble et al., Phys. Rev. C 28 (1983) 1552.
PHYSICS LETTERS B
2 July 1987
COALESCENCE-MODEL ANALYSIS OF ALPHA-PARTICLE AND DEUTERON SPECTRA FROM ENERGETIC HEAVY-ION COLLISIONS S. D A T T A , R. (~APLAR, N. C I N D R O Rudjer Bo2kovi6Institute, 41001 Zagreb, Croatia, Yugoslavia R.L. AUBLE, J.B. BALL A N D R.L. R O B I N S O N Oak Ridge National Laboratory, Oak Ridge, TN 37831, USA Received 2 January 1987; revised manuscript received 10 April 1987
Double differential cross sections for alpha particles and deuterons emitted from various colliding nuclear systems are analysed in terms of the coalescence model. The systems vary from light, symmetric to heavy, asymmetric ones; projectile energies are 15-25 MeV/nucleon. The obtained values of coalescence radii in momentum space show a striking regularity with A/(Aj,+AT).
The coalescence m o d e l o f complex-particle production in nuclear reactions has been successfully used at high b o m b a r d i n g energies. The m o d e l as form u l a t e d in refs. [ 1,2] accounts for the f o r m a t i o n o f high-energy deuterons from targets exposed to 2 5 - 3 0 GeV protons. The p r o p o s e d f o r m a t i o n m e c h a n i s m considers the pairing o f nucleons from cascades which develop in the struck nucleus. Basic to the idea is the concept o f coalescence radius [2] as the radius o f a small sphere a r o u n d the p o i n t representing any nucleon in m o m e n t u m space. Any n u m b e r o f nucleons whose relative m o m e n t u m is smaller than the coalescence radius forms a complex particle which is then emitted. Recently, several authors [ 3 - 5 ] have used the coalescence idea to explain the spectra o f complex particles, d, t, 3He, and ct's, e m i t t e d in heavy-ion reactions at energies a r o u n d 20 MeV/nucleon. This application, however, is not without conceptual and practical difficulties. In the present p a p e r we refine the coalescence model so as to make it m o r e a p p r o p r i a t e for use at energies o f 15-25 MeV/nucleon. In this way we obtain information on the systematic b e h a v i o r o f coalescence radii with system masses at these energies. We start by assuming that p r e - e q u i l i b r i u m emis302
sion o f nucleons in reactions induced by nucleons o f as low an energy as 15-25 MeV can be related to cascades initiated by these nucleons in the target nucleus. Thus, the coalescence-model ideas imply that the coalescence takes place among nucleons participating in the first few collisions, i.e., before the equilibrium sets in. In our t r e a t m e n t we correctly take into account the effects o f the C o u l o m b barrier by calculating the C o u l o m b repulsion between the e m i t t e d particles a n d the emitting system. Such a correction was i n t r o d u c e d by Awes et al. [3,4] as a fitting parameter. F u r t h e r m o r e , we properly treat the influence o f the nuclear-matter e n v i r o n m e n t on the formation o f the complex particle by introducing a mass correction which takes into account its b i n d i n g energy. The analysed d a t a consist o f double differential cross sections o f protons, deuterons, and ct particles e m i t t e d from the colliding systems listed in table 1, m e a s u r e d at l a b o r a t o r y angles o f 10 °, 30 °, 50 °, and 70 ° [6] #1 The derivation o f the model formulas used for data analysis follows the lines o f refs. [3,4]; we have, however, carried it out by including the complexparticle b i n d i n g energy; hence, now ma ¢:A.mp. The "~ Data were obtained at the Holifield Heavy Ion Research Facility, ORNL, Oak Ridge, TN.
0370-2693/87/$ 03.50 © Elsevier Science Publishers B.V. ( N o r t h - H o l l a n d Physics Publishing D i v i s i o n )
Volume
192, number
3,4
Table 1 Table of system- and angle-average System
PHYSICS
coalescence
LETTERS
B
2 July 1987
radii. Average coalescence
&,IA (MeV)
radius ( MeVlc)
angle (degrees)
system
10
30
50
70
alphas
j60+ “Al ‘hO+‘hTi ‘hO+6”Ni Ib0+ “OSn 160+ 19’Au “S + ‘,A, 3’S+ ?,A, 3ZS+ 4~Ti 32s + 46Ti “S + h”Ni 3’S+ “7AU 3’S+ “7AU S”Ni + h”Ni 58Ni + I‘“Sn SnNi+ “jSn 5”Ni + “‘Au
25.2 25.2 25.2 25.5 25.2 15.7 21.2 15.7 21.2 15.7 15.7 21.2 15.1 15.1 15.1 15.1
80 80 80 91 109 67 90 65 67 69 90 103 65 78 80 99
84 86 85 95 100 66 94 71 71 72 94 90 68 82 68 98
99 105 103 118 122 86 90 85 93 90 107 110 86 104 104 116
90 101 96 123 129 85 91 88 89 87 116 109 87 105 100 113
86 92 88 101 113 74 75 78 77 78 99 101 73 90 85 104
deuterons
“0 + 4hTi IhO+ 19’Au “S+ ‘07AU 58Ni + “ONi
25.2 25.2 21.2 15.1
64 90 87 46
68 87 81 48
87 112 103 71
81 125 105 69
73 101 92 57
cross section for the emission of complex particles of mass number A=N+ 2 can be obtained from the proton-emission spectra using the relation [ 7-101
with VZ.,~ the mass of the complex particle andf, Coulomb repulsion energy,
($),,=A (2)”
x4= 1.29(AT+Ap
(1) where N, Np, and NT. are, respectively, the neutron numbers of the complex particle, the projectile, and the target nucleus, and, similarly, 2, Zp, and 2, are the corresponding proton numbers. PO is the coalescence radius discussed earlier and otO, is the total cross section, for which we use the expression from ref. [ 111. As a next step, we assume that the complex particle is formed near the surface of the emitting nucleus. Then, denoting the momentum of the complex particle at the nuclear surface and when detected by P,4,, and PA, respectively, we have P.:/2m,d = P&J2m,4 +fA ,
(2)
1.44(Z,$Z,-Z)Z
the
MeV -A)“3
’
(3)
We also assume that the direction of the emitted particle is not affected by the Coulomb field. Then it can be shown that double differential cross sections for the emission of complex particles (d’ald&dQ), can be calculated from the measured proton cross sections (d’aldE,dS2),,,,,, by using the expression
with
Volume 192, number 3,4
PHYSICS LETTERSB
where p is the proton m o m e n t u m in the laboratory system, mp its mass, and fp the Coulomb barrier for protons [eq. (3) with A = Z = I ] . The relation between the laboratory energy AA of the complex particle A and that of the proton, Ep, is given by
2 July 1987
10"s
)0 "~
~ llD~i~llh'~ 0(3 "Vil~
x.~
E.4 = Azmp (Ep - f p ) +fro • mA
(6)
The importance of taking into account the effects of the Coulomb barrier and that of the particle binding energy, as done explicitly in eq. (6), can be readily recognized by considering the ratios EA/Ep of energies at which the observed complex-particle and proton spectra should be compared. This ratio influences expression (4) via the corresponding double differential cross sections which fall off exponentially with the energy of the emitted particle. For alpha particles, for instance, the ratio changes from E A / E p = 4 at very high energies (and taking mA = 4mp) to EA/Ep= 3-3.5 at spectral energies encountered in the present analysis and taking into account the correct mass m~= 4 m p - 2 6 . 7 MeV/c 2. This refinement considerably modifies the value of C given by (4), hence the values of Po extracted from (5). The analysis of the experimental data proceeds as follows. We first check the physical consistency of the values of the coalescence radius obtained for each system at various angles and energies. This we do by taking the ratio C as defined in expression (4), with the double differential cross sections representing the measured alpha (A = 4) and proton energy spectra, respectively; similarly, for deuterons. Then, using eq. (5) we extract the corresponding values of the coalescence radius Po in the following way. For a colliding system, we obtain the values of C from the spectra at various angles at energy intervals AEp = 1-2 MeV and the corresponding AE~=AE~ and AEA=AEd [see eq. (6)]. Thus a series of values of Po are obtained for each colliding system and each angle. The observation we make is that for the higherenergy part of the spectra, above some minimal energy, say, E min, the values obtained for P0 lie within _+5% of the average value for a given angle. At the most forward angle (10 ° ) this minimal energy lies well above 1.5-2 times the energy of beam-velocity alphas and deuterons. At 70 ° E ~ is 3-4 times above the corresponding values of the Coulomb barrier fA. In this way we obtain what we call the angle-average
coalescence radii. 304
SYSTEM
1
3ZS+ '97Au--'-X + d E,n{ = 6?9.8
li
O SYSIEM AVERAGE R DIUS X ANGLE AVERAGERADIUS • EXPERIMENT
-,,~
xx b ~
10.3
10.2
10,~
i
10 o
10-~
10-~
10-3
J 10
30
i
,
SO
10
?
;
~
90
I
h
110
~
. . . . . .
130
150
170
190
Ed (MeV)
Fig. 1. Deuteron spectra from 32S+ 197Auat E~nc=679.8 MeV. Finally, we take the statistically weighted average of all angle-average coalescence radii and obtain the coalescence radius for a given system at a given incident energy which we call the system-average coalescence radius. These radii are shown in the last column of table 1. Fig. 1 shows the experimental deuteron spectra for the 328i-k- 197Au system at 679.8 MeV and the spectra calculated by using (i) angle-average coalescence radii obtained, respectively, at 0~ab= 10 °, 30 °, 50 °, and 70 °, and (ii) the system-average radius. There is overall agreement with the experimental data, as well as quite satisfactory agreement between the two sets of calculated values, which demonstrates the consistency of our procedure. The agreement with
Volume 192, number 3,4
PHYSICS LETTERS B
140
120
~x~ ALPHAS
C~
Ap [SMALL AP+AT L LARGE
,-g
>~ lOO
~ %o -..~ ~'°-. ~ o "--a..
o') ¢0 e~ 0
tJ m
120" ~. %
0
~ 100
x%,
0 SYSTEM AVERAGE RADIUS
DEUTERONS
AVERAGES FOR: o ANGLES 10" AND 30' z~ ANGLES 50' AND 70'
80
60
40
0.1
0.2
0.3
0.4
0.5
0.6
0.7
Ap/(Ap+AO Fig. 2. Extracted coalescence radii Po versus Ap/(Ap+AT) for various colliding systems. the data is better for the high-energy part of the spectra, as expected. Similar results are obtained for deuteron and a spectra for all other studied systems. The values of the coalescence radii for alpha particles and deuterons obtained in the analysis are plotted versus Ap/(Ap +AT) in fig. 2. Two features of this figure deserve our attention. Firstly, the system-average values obtained for coalescence radii display a striking regularity with Ap/(Ap+AT): the coalescence radii increase with decreasing values of this parameter (larger Po for asymmetric colliding systems). The effect, seen for both alphas and deuterons, can be approximated by a parabolic best-fit curve (full lines). Secondly, as mentioned earlier, the angle-average coalescence radii obtained at more forward angles (10 ° and 30 °) are systematically smaller than those obtained at larger angles (50 ° and 70 ° ) for the same system. To point out this feature, in fig. 2 we have plotted separately arithmetic means of 10 ° and 30 ° and of 50 ° and 70 °
2 July 1987
radii for each analysed system. They maintain the parabolic trend of the system radii, but with two distinct parabolas (dashed lines). The decrease of Po with increasing Ap/(Ap+AT), i.e., when moving to more symmetric colliding systems, is expected from a crude picture of the process illustrated by the inset in fig. 2. In collisions of systems with large asymmetry [small values of Ar,/(Ap+AT)], the fraction of particles emitted in central collisions is expected to be larger than for symmetric systems. In the former, hence, a cascade nucleon finds a sufficient number of nucleons moving in its vicinity within a given direction to form a complex particle to be emitted. In symmetric systems, however, a cascade nucleon in the periphery will find a lesser number of nucleons suitable for coalescing and forming a complex particle, hence will require a larger volume radius (smaller momentum radius) to do so. Thus an effectively smaller coalescence radius Po for symmetric Systems. It is worth noting that in the studied Ei,c/A range (15-25 MeV/nucleon) system coalescence radii obtained from the model display little, if any, energy dependence. The deuteron radii P0 obtained from the analysis compare well with values reported in the literature; the alpha radii, however, are somewhat smaller [ 12 ]. A thermodynamic calculation based on ref. [10] gives the radius R of a thermalized volume corresponding to the momentum-space coalescence radius P0 as
R = { [Z!N~.A3(2s+1 )exp(Eo/T)]/2 A } ~/3(A ~) × (3/4n) 2/3h/Po ,
(7)
where Eo is the binding energy of the composite particle and s its spin. Using T = 11 MeV (ref. [ 13]) and a mean v a l u e / o = 9 0 MeV/c, we obtain R~---9.5 fm for alpha particles. Similarly, with Po = 80 MeV/c, we obtain Rd= 11.6 fm for deuterons. The latter value compares well with that reported in ref. [ 14]. The meaningfulness of such large spatial radii, encompassing essentially the whole nucleus, should be questioned and, with it, the applicability of eq. (7) to the energy range E / A < 100 MeV/u. The uncertainty principle defines the spatial coalescencesphere radius as
305
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R <~h/Po ,
PHYSICS LETTERS B (8)
yielding ~ 2 . 2 fm for alphas a n d ~ 2 . 5 fm for deuterons. Such radii are more consistent with what we know about the n u c l e o n - n u c l e o n interaction. Concluding, we should say that (i) we have refined the coalescence model by taking into account the b i n d i n g energy in the masses of the complex ejectiles and applied it to the analysis of alpha a n d deuteron spectra from heavy-ion collisions. This correction a n d the properly handled Coulomb-barrier correction have a considerable influence on the extracted values of the coalescence radii Po; (ii) the extracted values of Po show a striking dependence on Ap/(Ap+AT), increasing regularly from symmetric to asymmetric colliding systems. This dependence is compatible with a geometrical picture of the reaction mechanism. This paper is part of a study u n d e r the US-Yugoslav collaboration project JFP 554 DOE/IRB. The financial support of the j o i n t
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US-Yugoslav Board for Scientific a n d Technological Cooperation is gratefully acknowledged.
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