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Kinetic and internal fragment temperatures on a highly excited nucleus
Volume 228, number 4
PHYSICSLETTERSB
28 September 1989
KINETIC AND INTERNAL FRAGMENT TEMPERATURES OF A HIGHLY EXCITED NUCLEUS H.W. B A R Z a, j.p. B O N D O R F h, R. D O N A N G E L O
c, H. S C H U L Z a and K. S N E P P E N b
"~ ZfK Rossendorf DDR-8051 Dresden, GDR Niels Bohr Institute, Blegdamsvej 17, DK-2100 Copenhagen, Denmark Instituto de Fisica da UFRJ, 21945 Rio de Janeiro, Brazil Received 13 June 1989
The kinetic and internal temperatures of the fragments produced in a nuclear break-up process are calculated in a VUU plus cluster emission treatment. The results are in accordancewith the experimentally measured values.
Fragmentation processes have been studied quite successfully on the basis of statistical models (see refs. [ 1-3 ] ). They appcar to give an adequate description of the multiplicity, of both the mass number and isotopic distributions, and, in particular, of the excitation spectra of the fragments observed in the exit channel of such reactions. However, there is still a problem which cannot be solved neither within these nor any other non-dynamical models we are aware of, namely, that of the observation of two distinct temperature values (see e.g. ref. [4] ). One is the kinetic temperature 7kin, which is determined from the fragments' energy spcctra, and the other is the internal temperature Trot, which is extracted from the population of particle unbound states of single fragments. In particular, 7~,, is relatively insensitive to both the projectile nucleus and to the beam energy in the range of 35-94 MeV per nucleon (for a recent review see ref. [ 5 ] ). We expect to understand this problem only in the context of a full dynamical treatment which includes correlations and other many-body aspects of the collision process. Progress in this direction has recently been made by using semiclassical mcthods such as Vlasov-Uehling-Uhlenbeck (VUU) [6 ] or thc molecular dynamical approaches [ 7 ]. However the description of the formation of the observed quantum states (nuclear clusters with internal excitation) lies outside the scope of these models which are essentially based on classical distribution functions. In order to gain more insight into the dynamics of
fragment formation in a heavy ion reaction we consider in the present letter two complementary aspects. These are, the evolution of the system from the initial stages of the collision until its maximum expansion, and its subsequent break-up into fragments. During the first part of the reaction the mean field including the collision term dominates the dynamics and the nucleus-nucleus collision can adequately be treated within the Vlasov-Uehling-Uhlenbeck approach. This formalism is, however, unable of describing the collapse of the expanded nuclear system into fragmcnts. To treat this last process we could have estimated the emission of clusters using some of the currently available statistical multifragmentation models [8]. Instead, we have utilized Weisskopf's [9] evaporation model for an expanding nuclcus. This model will give us at least a qualitative understanding of the time evolution of the nucleus towards its stage of disassembly by calculating the corresponding phase space for cluster formation. In the VUU approach the excited nucleus is described through the one-body distribution function f ( x , p, t) which depends on the coordinate and the momentum of a nucleon. The time evolution of this function is governed by the VUU equation:
0370-2693/89/$ 03.50 ©Elsevicr Science Publishers B.V. ( North-Holland Physics Publishing Division )
~+
p Of
OUOJ'=I
m 0x
0x ~p
(I) '
where the mean field potential U is a given function of the spatial density satisfying the saturation condition of nuclear matter. The quantity I stands for the 453
Volume 228, number 4
PI-IYSICS LETTERS B
collision integral. This equation is solved using thc code described in ref. [ 10]. For simplicity, instead of considcring the full collision process, we take as initial condition for the VUU calculation the distribution function o f a thermalized Fermi-gas:
excited nucleus having initially 40 nucleons and an excitation energy of E/A = 15 MeV is shown in fig. la. We see that within the first 30 fm/c the system loses more than half of its nucleons, while its density drops to roughly one third of the normal nuclear matter density Po. The reason for this is that nucleons whose kinetic energy exceeds the potential barrier are rapidly emitted, l f w c proceeded with this VUU description, at later times the density would recover and the residual nucleus would undergo damped oscillations as indicated in fig. la. However, as we have stressed, in this semiclassical VUU approach genuine cluster formation cannot occur and only particles can be emitted. We expect that the formation of clusters will take place predominantly when the expanding nucleus is stressed to its extreme. At this point we give up thc VUU description and utilize a model based on Weisskopf's evaporation picture [9]. The evaporation model enables us to estimate the emission rates of both nucleons and clusters on the basis of quantum mechanics. For the sake ofsimplic-
fix, p,/=0)= (2/rmT) -3/2 __pt
2
i.e. we omit in our calculation the compression stage of the collision. Fig. 5b in ref. [ 10] illustrates the full time evolution, starting before the participant nuclei entered in contact. Our initial condition would thus roughly correspond to time t ~ 60 fm/c in that figure. The temperature T is connected with the excitation energy E via T = {E/A, where A is the nuclear mass number. The quantityf~(x, p) stands for the distribution function of the cold nucleus as its saturation density. The evolution of the density and the mass of an
i0 m r
k
r
-
r
-
i
_~
"\
\
20 -
1
\
_J I
~
u
o_<
0 V ....... 0.15
k
I"\
q.
0.10 ~
~ -
]"
L
oi
i
.... X:-.'.; A°., [__
I-
I
density 0!5
12
010
8,-,
#3
'E
\\
/
o.. ,4
005
0.05 [
i ,
O/ C
r
°
I0 L
~ ,
Ac= 7
| mGss number
\
30
-
28 September 1989
I
I00 t (fro/c)
L. . . . . .
[.--
200
o
o
I00
1__ 0
2O0
t (fro/c)
Fig. 1. (a) Time evolution for mass number and density of a hot nucleus with an excitation energy of 15 McV per nucleon and A = 40. The dashed lines show the results of the VUU approach. The solid curves are calculated by utilizing Weisskopf's evaporation picture startcd at the point on the V U U path where the nucleus is most stressed. (b) Time evolution of the hot nucleus (see ( a ) ) but calculated with Weisskopf's evaporation model coupled to an expanding nucleus. Lower panel: density and temperature evolution; upper panel: emission probability A,J°,~,. (arbitrary units ) tbr fragments of various masses.
454
Volume 228, number 4
PHYSICS LETTERS B
ity we consider for the moment the evolving nucleus of mass number A as a fluid droplet of radius R, density p = 3A/4nR 3 and temperature 7~ i.c. in generalizing Weisskopf's original picture we consider now lhe emission process under variations of the density of the emitting source. For that purpose we are using a velocity field which is proportional to the distance from the center of the droplet. The equation of motion is then simply a function of the droplet's radius R:
3mA ( R ' - rl[~) = 4~zR 2p_ 8nRa,
(3)
where m, p, r/, a are the nucleon mass, pressure, friction and surface tension coefficients, respectively. The evaporation of fragments into various channels c enters via the energy balance equation
d (eR3)+ p d R3
d--t
/
= - 3tlmAR2- ~. ac explq-. (F (A, p, T) \1 5:-F(A-A,.,p, T)-C.)),
(4)
where e is the encrgy density. The emission rate depends strongly on density and temperature via the free energy F and the binding energy - E , and mass number Ac of the emitted cluster. The exponential factor comes from the fact that the emission probabilities are proportional to the level density of the residual nucleus and inversely proportional to that of the decaying nucleus. The kinetic energy of the emitted particle, the corresponding Coulomb energy and barrier and the compound nucleus formation cross section are contained in the quantity ac (see ref. [ 11 ] ). An important property of eq. (4) is that at low density the specific free energy is small and the cluster binding energy -E,. plays the decisive role. Therefore the emission of composite particles increases strongly when the matter becomes more and more diluted [12]. This can be seen from fig. lb, where in the lower part the density and temperature cvolution is shown, while the upper part of this figure gives the probability for cluster emission as a function of time. One sees clearly that cluster emission increases strongly when the density of the expanding system approaches p.w.Po/3 and the temperature T ~ 6
28 September 1989
MeV. It is also interesting to observe that the fragments with Ac= 3 are initially evaporated as it would follow from the usual Weisskopfevaporation picture (constant density) but as soon as the density gets rather diluted the gain in binding energy for the 3H and 3He nuclei becomes essential and this way the emission probability gets also peaked around a temperature of 6 MeV. These results suggest that the Weisskopf picture allows us to at least qualitatively include fragment formation in the VUU approach. We therefore consider the cluster emission process at the point on the VUU path at which the expanding nuclcus is stretched to its extreme (p~po/3, 7"~ 8.5 MeV). The decay of the nucleus now takes place in accordance to the emission probabilities shown in fig. lb. One sees that the density of the expanding blob does not change very much (see also fig. lb) and evolves very similar to that of the VUU calculation suggesting a break-up density of about p~po/3. The emission temperature is in accordancc to the results displayed in fig. lb and amounts to the order of 6 MeV. The mass number decreases rapidly with time (fig. la), but this cffcct would still be seen in a better way when considering a much bigger source (A >7 100). In this case the corresponding VUU calculation becomes very time consuming and was, for this reason, not performed. At this point the following comment is in order. The simplified scenario of an expanding fluid blob (eqs. (3), ( 4 ) ) leads only to an emission temperature of about 5-6 MeV when the initial heating or comprcssion of the system is high enough so that the systcm always cxpands. Moreover, for relatively small systems the term in eq. (3) associated with the surface energy significantly slows down the expansion and gives rise to an oscillating behaviour of the density. Thus a description based solely on an expanding fluid blob would completely fail to describe the fragment emission. However, the combination of the VUU approach with Weisskopf's evaporation model we have introduced hcrc gives an emission temperature of the order of 5-6 MeV, even for moderate excitations. This way one can at least qualitatively understand the experimentally given facts showing that the internal temperature T,n, of the formed fragments is relatively insensitive to both the projectile nucleus and to the beam energy in the range of 35-94 MeV per nucleon (see ref. [5]). This remarkable con455
Volume 228, number 4
PHYSICS LETTERS B
stancy of 7,n t suggests that there may be a limiting temperature of about 5-6 MeV above which fragment formation is strongly suppressed. This is also in agreement with the predictions of the above mentioned statistical models (see refs. [ 1-3 ] ). The internal temperature T~,, is, howevcr, usually much lower than the kinetic one Tk~,, which is obtained by comparing the energy spectra of the fragments with that o f a thermalized moving source. One gets "lkm~ 10--25 MeV from the above mentioned expcriments (scc ref. [5 ] ). Dcspite of several appcaling features Weisskopf's evaporation model probes the clusterization probability locally under the assumption of thermodynamical equilibrium. It may therefore give unreliable rcsults when calculating the fragments energy spcctra which contain the imprint of the time evolution of the emitting source. In this respect one faces the same kind of difficulties inherent in the statistical multifragmentation models (refs. [1-3 ] ), because dynamical effects during the clusterization stage are not included (for a study of the Coulomb expansion we refer to ref. [ 13 ] ). To understand the nature of these two distinct temperatures in a way that takes into account the full time evolution of the system, i.e. clusterization and expansion, let us for a moment replace the actual VUU distribution function by the following idealizcd one,
f(r,p, t)oc exp
p~ 3(r-pt/m)2"~ 2mTo 2R~ / .
(5)
28 September 1989
has to be associated with the local one at which the bound state formation is most probable. To get a handle on these two distinct temperatures we employ the coalescence model to calculate the probabilities for forming a cluster in its ground and excited state. In ref. [4 ] the distribution function (5) was used. Here we will employ the more appropriate distribution function which emerges from the solution of the VUU equation ( 1 ) and contains the modifications duc to the mean field as well as the collision term. For simplicity we consider, as in ref. [4], the sudden formation of a two-nucleon cluster having one excited state (quasi-deuteron). The internal temperature is derived from the ratio of the occupation probabilities of the ground and excited state of the quasi-deuteron. In fig. 2 we show the kinetic and internal temperatures as function of the expansion time. The initial excitation energy per nuclcon is again 15 MeV. One secs that the resulting kinetic temperature is Tkin~ 10 MeV, while the internal temperature of the cluster is significantly smaller than Tki. and changes with time, i.e. 7~,, is a function of the density of the system at the time of break-up. We have also pcrformed similar calculations with four-body clusters, which, although they suffer from large statistical errors due to the small number of test particles, seem to confirm the trend illustratcd in fig. 2. Modelling the cluster formation within the Weisskopf picture is much simpler and gives at the same time a qualitatively satisfactory insight into the rather complicated and involved mech-
As the system expands the local temperature of a comoving volume decreases adiabatically, so that T 3~o~.:,t / 2 =const. However, the single-particle spectrum of the nucleons moving towards the detectors are governed by the initial temperature 7o and not by the local one, since
p~
p2
dNfdrf(r'p't)'~exp( 2---~o) dp
(6,
is independent of time. Thus the source temperature that must be associated with the temperature Tk~, extracted from the moving source fits is 70, and not some time average of the system temperature, as it could perhaps be thought. If composite particles are formed then their energy spectra are also governed by 7o according to (6), while the internal temperature 456
I--
5
i o~ o
,__
I 5o
,
1 .., ~oo
I .... ~5o
J. . . . . . 2oo
2~o
t [fm/c]
Fig. 2. The internal and kinetic temperature of quasi-deuterons obtained from VUU calculations as a function of the expansion time. The kinetic temperaturcs (triangles) are calculated from the spectra and fitted by the dashed line.
Volume 228, number 4
PHYSICS LETTERS B
anism of fragment formation, although certain dynamical aspects are still poorly taken into account. In summary, wc have studied the time evolution of an excited nucleus by making use of the V U U approach. To investigate the cluster formation we utilized Weisskopf's evaporation picture for an cxpanded nucleus. We found that in V U U type calculations the intermediate mass formation should occur p r e d o m i n a n t l y when the local temperature is about 6 MeV and the density p~po/3. To reconcile this internal temperature of the fragments, which characterizes the temperature at which the system breaks up and which is significantly smaller than the one determined from the slopc of the fragment's energy spectra, we used a coalescence model together with the V U U approach. This simplified consideration reflects the experimental finding, that the fragments have an inlernal temperature of about 6 MeV, while their kinetic temperature is much higher and scales mainly with the beam energy. H.W.B., R.D. and H.S. thank the Niels Bohr Instilute for hospitality and partial financial support. R.D. also acknowledges support from the CN Pq (Brazil).
28 September 1989
References [ 1] S.E. Koonin and J. Randrup, Nucl. Phys. A 356 ( 1981 ) 223; A484 (1987) 173. [2] J.P. Bondorf, R. Donangelo, I.N. Mishustin, C.J. Pethick, H. Schulzand K. Sneppen, Nucl. Phys. A 443 ( 1985 ) 321 ; J.P. Bondort, R. Donangelo,I.N. Mishustin, and H. Schulz, Nucl. Phys. A 444 (1985) 460. [3] B.-H. Sa and D.H.E. Gross, Nucl. Phys. A 437 (1985) 643; X.Z. Zhang, D.H.E. Gross and Y.M. Zheng, Nucl. Phys. A 461 (1987) 641;A461 (1987) 668. [4] H.W. Barz, H. Schulz and G.F. Bertsch, Phys. Lett. B 217 (1989) 397. [5] C.K. Gelbke and D.H. Boal, Prog. Part. Nucl. Phys. 19 (1987) 33. [6] G.F. Bertsch and S. Das Gupta, Phys. Rep. 160 (1988) 189, and referencestherein. [7] J. Aichelin et al., Phys. Rev. C 37 (1988) 2451. [ 8 ] K. Sneppen and L. Vinet, Nucl. Phys. A 480 (1988) 342. [9 ] V.F. Weisskopf, Phys. Rev. 52 (1987) 295. [10] C. Gregoire et al., Nucl. Phys. A 465 (1987) 317. [ 11 ] W.A. Friedman and W.G. Lynch, Phys. Rev. C 28 (1983) 16. [ 12 ] W.A. Friedman, Phys. Rev. Len. 60 ( 1988) 2125. [ 13] H.W. Bar'z, J.P. Bondorf and H. Schulz, Nucl. Phys. A 462 ( 1987 ) 742.
457
PHYSICSLETTERSB
28 September 1989
KINETIC AND INTERNAL FRAGMENT TEMPERATURES OF A HIGHLY EXCITED NUCLEUS H.W. B A R Z a, j.p. B O N D O R F h, R. D O N A N G E L O
c, H. S C H U L Z a and K. S N E P P E N b
"~ ZfK Rossendorf DDR-8051 Dresden, GDR Niels Bohr Institute, Blegdamsvej 17, DK-2100 Copenhagen, Denmark Instituto de Fisica da UFRJ, 21945 Rio de Janeiro, Brazil Received 13 June 1989
The kinetic and internal temperatures of the fragments produced in a nuclear break-up process are calculated in a VUU plus cluster emission treatment. The results are in accordancewith the experimentally measured values.
Fragmentation processes have been studied quite successfully on the basis of statistical models (see refs. [ 1-3 ] ). They appcar to give an adequate description of the multiplicity, of both the mass number and isotopic distributions, and, in particular, of the excitation spectra of the fragments observed in the exit channel of such reactions. However, there is still a problem which cannot be solved neither within these nor any other non-dynamical models we are aware of, namely, that of the observation of two distinct temperature values (see e.g. ref. [4] ). One is the kinetic temperature 7kin, which is determined from the fragments' energy spcctra, and the other is the internal temperature Trot, which is extracted from the population of particle unbound states of single fragments. In particular, 7~,, is relatively insensitive to both the projectile nucleus and to the beam energy in the range of 35-94 MeV per nucleon (for a recent review see ref. [ 5 ] ). We expect to understand this problem only in the context of a full dynamical treatment which includes correlations and other many-body aspects of the collision process. Progress in this direction has recently been made by using semiclassical mcthods such as Vlasov-Uehling-Uhlenbeck (VUU) [6 ] or thc molecular dynamical approaches [ 7 ]. However the description of the formation of the observed quantum states (nuclear clusters with internal excitation) lies outside the scope of these models which are essentially based on classical distribution functions. In order to gain more insight into the dynamics of
fragment formation in a heavy ion reaction we consider in the present letter two complementary aspects. These are, the evolution of the system from the initial stages of the collision until its maximum expansion, and its subsequent break-up into fragments. During the first part of the reaction the mean field including the collision term dominates the dynamics and the nucleus-nucleus collision can adequately be treated within the Vlasov-Uehling-Uhlenbeck approach. This formalism is, however, unable of describing the collapse of the expanded nuclear system into fragmcnts. To treat this last process we could have estimated the emission of clusters using some of the currently available statistical multifragmentation models [8]. Instead, we have utilized Weisskopf's [9] evaporation model for an expanding nuclcus. This model will give us at least a qualitative understanding of the time evolution of the nucleus towards its stage of disassembly by calculating the corresponding phase space for cluster formation. In the VUU approach the excited nucleus is described through the one-body distribution function f ( x , p, t) which depends on the coordinate and the momentum of a nucleon. The time evolution of this function is governed by the VUU equation:
0370-2693/89/$ 03.50 ©Elsevicr Science Publishers B.V. ( North-Holland Physics Publishing Division )
~+
p Of
OUOJ'=I
m 0x
0x ~p
(I) '
where the mean field potential U is a given function of the spatial density satisfying the saturation condition of nuclear matter. The quantity I stands for the 453
Volume 228, number 4
PI-IYSICS LETTERS B
collision integral. This equation is solved using thc code described in ref. [ 10]. For simplicity, instead of considcring the full collision process, we take as initial condition for the VUU calculation the distribution function o f a thermalized Fermi-gas:
excited nucleus having initially 40 nucleons and an excitation energy of E/A = 15 MeV is shown in fig. la. We see that within the first 30 fm/c the system loses more than half of its nucleons, while its density drops to roughly one third of the normal nuclear matter density Po. The reason for this is that nucleons whose kinetic energy exceeds the potential barrier are rapidly emitted, l f w c proceeded with this VUU description, at later times the density would recover and the residual nucleus would undergo damped oscillations as indicated in fig. la. However, as we have stressed, in this semiclassical VUU approach genuine cluster formation cannot occur and only particles can be emitted. We expect that the formation of clusters will take place predominantly when the expanding nucleus is stressed to its extreme. At this point we give up thc VUU description and utilize a model based on Weisskopf's evaporation picture [9]. The evaporation model enables us to estimate the emission rates of both nucleons and clusters on the basis of quantum mechanics. For the sake ofsimplic-
fix, p,/=0)= (2/rmT) -3/2 __pt
2
i.e. we omit in our calculation the compression stage of the collision. Fig. 5b in ref. [ 10] illustrates the full time evolution, starting before the participant nuclei entered in contact. Our initial condition would thus roughly correspond to time t ~ 60 fm/c in that figure. The temperature T is connected with the excitation energy E via T = {E/A, where A is the nuclear mass number. The quantityf~(x, p) stands for the distribution function of the cold nucleus as its saturation density. The evolution of the density and the mass of an
i0 m r
k
r
-
r
-
i
_~
"\
\
20 -
1
\
_J I
~
u
o_<
0 V ....... 0.15
k
I"\
q.
0.10 ~
~ -
]"
L
oi
i
.... X:-.'.; A°., [__
I-
I
density 0!5
12
010
8,-,
#3
'E
\\
/
o.. ,4
005
0.05 [
i ,
O/ C
r
°
I0 L
~ ,
Ac= 7
| mGss number
\
30
-
28 September 1989
I
I00 t (fro/c)
L. . . . . .
[.--
200
o
o
I00
1__ 0
2O0
t (fro/c)
Fig. 1. (a) Time evolution for mass number and density of a hot nucleus with an excitation energy of 15 McV per nucleon and A = 40. The dashed lines show the results of the VUU approach. The solid curves are calculated by utilizing Weisskopf's evaporation picture startcd at the point on the V U U path where the nucleus is most stressed. (b) Time evolution of the hot nucleus (see ( a ) ) but calculated with Weisskopf's evaporation model coupled to an expanding nucleus. Lower panel: density and temperature evolution; upper panel: emission probability A,J°,~,. (arbitrary units ) tbr fragments of various masses.
454
Volume 228, number 4
PHYSICS LETTERS B
ity we consider for the moment the evolving nucleus of mass number A as a fluid droplet of radius R, density p = 3A/4nR 3 and temperature 7~ i.c. in generalizing Weisskopf's original picture we consider now lhe emission process under variations of the density of the emitting source. For that purpose we are using a velocity field which is proportional to the distance from the center of the droplet. The equation of motion is then simply a function of the droplet's radius R:
3mA ( R ' - rl[~) = 4~zR 2p_ 8nRa,
(3)
where m, p, r/, a are the nucleon mass, pressure, friction and surface tension coefficients, respectively. The evaporation of fragments into various channels c enters via the energy balance equation
d (eR3)+ p d R3
d--t
/
= - 3tlmAR2- ~. ac explq-. (F (A, p, T) \1 5:-F(A-A,.,p, T)-C.)),
(4)
where e is the encrgy density. The emission rate depends strongly on density and temperature via the free energy F and the binding energy - E , and mass number Ac of the emitted cluster. The exponential factor comes from the fact that the emission probabilities are proportional to the level density of the residual nucleus and inversely proportional to that of the decaying nucleus. The kinetic energy of the emitted particle, the corresponding Coulomb energy and barrier and the compound nucleus formation cross section are contained in the quantity ac (see ref. [ 11 ] ). An important property of eq. (4) is that at low density the specific free energy is small and the cluster binding energy -E,. plays the decisive role. Therefore the emission of composite particles increases strongly when the matter becomes more and more diluted [12]. This can be seen from fig. lb, where in the lower part the density and temperature cvolution is shown, while the upper part of this figure gives the probability for cluster emission as a function of time. One sees clearly that cluster emission increases strongly when the density of the expanding system approaches p.w.Po/3 and the temperature T ~ 6
28 September 1989
MeV. It is also interesting to observe that the fragments with Ac= 3 are initially evaporated as it would follow from the usual Weisskopfevaporation picture (constant density) but as soon as the density gets rather diluted the gain in binding energy for the 3H and 3He nuclei becomes essential and this way the emission probability gets also peaked around a temperature of 6 MeV. These results suggest that the Weisskopf picture allows us to at least qualitatively include fragment formation in the VUU approach. We therefore consider the cluster emission process at the point on the VUU path at which the expanding nuclcus is stretched to its extreme (p~po/3, 7"~ 8.5 MeV). The decay of the nucleus now takes place in accordance to the emission probabilities shown in fig. lb. One sees that the density of the expanding blob does not change very much (see also fig. lb) and evolves very similar to that of the VUU calculation suggesting a break-up density of about p~po/3. The emission temperature is in accordancc to the results displayed in fig. lb and amounts to the order of 6 MeV. The mass number decreases rapidly with time (fig. la), but this cffcct would still be seen in a better way when considering a much bigger source (A >7 100). In this case the corresponding VUU calculation becomes very time consuming and was, for this reason, not performed. At this point the following comment is in order. The simplified scenario of an expanding fluid blob (eqs. (3), ( 4 ) ) leads only to an emission temperature of about 5-6 MeV when the initial heating or comprcssion of the system is high enough so that the systcm always cxpands. Moreover, for relatively small systems the term in eq. (3) associated with the surface energy significantly slows down the expansion and gives rise to an oscillating behaviour of the density. Thus a description based solely on an expanding fluid blob would completely fail to describe the fragment emission. However, the combination of the VUU approach with Weisskopf's evaporation model we have introduced hcrc gives an emission temperature of the order of 5-6 MeV, even for moderate excitations. This way one can at least qualitatively understand the experimentally given facts showing that the internal temperature T,n, of the formed fragments is relatively insensitive to both the projectile nucleus and to the beam energy in the range of 35-94 MeV per nucleon (see ref. [5]). This remarkable con455
Volume 228, number 4
PHYSICS LETTERS B
stancy of 7,n t suggests that there may be a limiting temperature of about 5-6 MeV above which fragment formation is strongly suppressed. This is also in agreement with the predictions of the above mentioned statistical models (see refs. [ 1-3 ] ). The internal temperature T~,, is, howevcr, usually much lower than the kinetic one Tk~,, which is obtained by comparing the energy spectra of the fragments with that o f a thermalized moving source. One gets "lkm~ 10--25 MeV from the above mentioned expcriments (scc ref. [5 ] ). Dcspite of several appcaling features Weisskopf's evaporation model probes the clusterization probability locally under the assumption of thermodynamical equilibrium. It may therefore give unreliable rcsults when calculating the fragments energy spcctra which contain the imprint of the time evolution of the emitting source. In this respect one faces the same kind of difficulties inherent in the statistical multifragmentation models (refs. [1-3 ] ), because dynamical effects during the clusterization stage are not included (for a study of the Coulomb expansion we refer to ref. [ 13 ] ). To understand the nature of these two distinct temperatures in a way that takes into account the full time evolution of the system, i.e. clusterization and expansion, let us for a moment replace the actual VUU distribution function by the following idealizcd one,
f(r,p, t)oc exp
p~ 3(r-pt/m)2"~ 2mTo 2R~ / .
(5)
28 September 1989
has to be associated with the local one at which the bound state formation is most probable. To get a handle on these two distinct temperatures we employ the coalescence model to calculate the probabilities for forming a cluster in its ground and excited state. In ref. [4 ] the distribution function (5) was used. Here we will employ the more appropriate distribution function which emerges from the solution of the VUU equation ( 1 ) and contains the modifications duc to the mean field as well as the collision term. For simplicity we consider, as in ref. [4], the sudden formation of a two-nucleon cluster having one excited state (quasi-deuteron). The internal temperature is derived from the ratio of the occupation probabilities of the ground and excited state of the quasi-deuteron. In fig. 2 we show the kinetic and internal temperatures as function of the expansion time. The initial excitation energy per nuclcon is again 15 MeV. One secs that the resulting kinetic temperature is Tkin~ 10 MeV, while the internal temperature of the cluster is significantly smaller than Tki. and changes with time, i.e. 7~,, is a function of the density of the system at the time of break-up. We have also pcrformed similar calculations with four-body clusters, which, although they suffer from large statistical errors due to the small number of test particles, seem to confirm the trend illustratcd in fig. 2. Modelling the cluster formation within the Weisskopf picture is much simpler and gives at the same time a qualitatively satisfactory insight into the rather complicated and involved mech-
As the system expands the local temperature of a comoving volume decreases adiabatically, so that T 3~o~.:,t / 2 =const. However, the single-particle spectrum of the nucleons moving towards the detectors are governed by the initial temperature 7o and not by the local one, since
p~
p2
dNfdrf(r'p't)'~exp( 2---~o) dp
(6,
is independent of time. Thus the source temperature that must be associated with the temperature Tk~, extracted from the moving source fits is 70, and not some time average of the system temperature, as it could perhaps be thought. If composite particles are formed then their energy spectra are also governed by 7o according to (6), while the internal temperature 456
I--
5
i o~ o
,__
I 5o
,
1 .., ~oo
I .... ~5o
J. . . . . . 2oo
2~o
t [fm/c]
Fig. 2. The internal and kinetic temperature of quasi-deuterons obtained from VUU calculations as a function of the expansion time. The kinetic temperaturcs (triangles) are calculated from the spectra and fitted by the dashed line.
Volume 228, number 4
PHYSICS LETTERS B
anism of fragment formation, although certain dynamical aspects are still poorly taken into account. In summary, wc have studied the time evolution of an excited nucleus by making use of the V U U approach. To investigate the cluster formation we utilized Weisskopf's evaporation picture for an cxpanded nucleus. We found that in V U U type calculations the intermediate mass formation should occur p r e d o m i n a n t l y when the local temperature is about 6 MeV and the density p~po/3. To reconcile this internal temperature of the fragments, which characterizes the temperature at which the system breaks up and which is significantly smaller than the one determined from the slopc of the fragment's energy spectra, we used a coalescence model together with the V U U approach. This simplified consideration reflects the experimental finding, that the fragments have an inlernal temperature of about 6 MeV, while their kinetic temperature is much higher and scales mainly with the beam energy. H.W.B., R.D. and H.S. thank the Niels Bohr Instilute for hospitality and partial financial support. R.D. also acknowledges support from the CN Pq (Brazil).
28 September 1989
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