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Three-step analytic model for high-energy heavy-ion collisions
Nuclear Physics A415 (1984) 530-541 @ North-Holland Publishing Company
THREE-STEP ANALYTIC MODEL FOR HIGH-ENERGY HEAVY-ION COLLISIONS N.L. BALAZS”,
J.P. BONDORF b, B. LUKACS’
International Workshop for Theoretical Physics, H-1525
and J. ZIMANYI’
Budapest 114, POB 49, Hungary
Received 25 July 1983 Abstract: We discuss within the framework of an analytical model central collisions between large nuclei at an intermediate energy. The model assumes three stages: the ignition stage in which thermalized nucleons are created within the reaction volume; the expansion phase in which the hot nuclear matter expands and cools; and finally, the evaporation stage in which the system disintegrates through a surface evaporation. The model provides a qualitative insight into the connections between the main physical processes throughout the collision event.
1. Introduction Models of increasing sophistication have been suggested for describing high-energy heavy-ion collisions. However, for such models [e.g. cascade, relativistic hydrodynamics, hadrochemistry, rows on rows, volume evaporation etc., see e.g. refs. ‘-97”)] a great deal of numerical work is required. On account of the complexity of these calculations it is difficult to gain an insight into the basic physics of the collisions. Similarly, simple and calculable models are desirable as inputs for QCD calculations in order to enable one to perform sensible estimations of phenomena connected with the formation of quark-gluon plasma. For these reasons we have developed a simple analytic model describing the whole evolution of the system in a central collision between two big nuclei. In the spirit of ref. ‘) the process is divided into three essentially different stages, viz. ignition, expansion and break-up; however, in the present model the idea of a continuous evaporation is also introduced. This model will then be applied to clarify some problems connected with the blast-wave model lo). In presenting our model we are particularly concerned with advanced dimensional analysis using simple but fundamental ideas of physics such as the conservation laws and basic kinetic theory. We hope that in doing so we provide a sufficiently adequate caricature of the actual features of the physical process. Our approach is basically Permanent addresses: “) Department of Physics, State University of New York at Stony Brook, Stony Brook, NY 11794, USA. ‘) The Niels Bohr Institute, Blegdamsvej 17, DK-2100 Copenhagen 0, Denmark. ‘) Central Research Institute for Physics, H-1525 Budapest 114, POB 49, Hungary. 530
N.L. Balks
et al. / Three-step analytic model
531
as follows: During the evolution of the system we distinguish three stages; these are (i) the ignition, (ii) the expansion, (iii) the break-up by evaporation. The description of stages (i) and (ii) follows traditional ideas; however, these are arranged in such a manner that the process can actually be described by simple expressions. In stage (iii) we depart from the standard approach and characterize the usual breakup stage differently. We do this by recognizing the important role of the mean free path in limiting the validity of the hydrodynamic description in an inhomogeneous and finite system. Because of this, we shall treat stage (i) in a rather schematic fashion, its whole purpose being to provide initial data for stages (ii) and (iii) in which the role of the mean free path and the idea of a continuous evaporation will be displayed. The latter has, in particular, the important advantage that it replaces the rather ad hoc introduction of a sudden break-up by the more continuous process of evaporation, eliminating thereby the sharp division between stages (ii) and (iii).
2. Description 2.1. IGNITION
of the model
STAGE
Initially the system consists of two equal nuclei having sharp edges and being in contact; after this, the ignition stage starts; it ends when the two nuclei overlap completely. During this stage the velocity ‘distribution is becoming thermalized. Even though the detailed mechanism of this process may be quite complicated, we shall construct a simple and analytical model. As in ref. ‘) we do this by using some more or less technical approximations: (a) The compression is neglected. (b) Only nucleons are included, other particles such as r, A, etc. are not treated explicitly. This simplification is justified because, according to the hadron chemistry calculation of ref. ‘), the relative density of A’s never exceeds 10% in the considered energy range. (c) The particles are considered thermalized after the first collision. (d) Within the reaction volume (i.e. in the overlap region) density gradients are neglected. Although these simplifications might be considered substantial, it is easy to see that they are necessary: for example, (c) is required by our wish that the further phases of the evolution should be calculable assuming thermal equilibrium. Using item (a) one obtains for the reaction volume
where V, = $rRi is the nulcear volume before the collision, and v is the velocity of the nuclei in the c.m. system, v =$vlab.
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N.L. Baldzs et al. / Three-step analytic model
Items (b) and (c) mean that we distinguish only three different components of which two are cold, and one is thermalized. Because of the right-left symmetry only two different densities occur in the balance equations. The assumed reactions are as follows: N o.left+ No,right+ N + N 9 N O,(leftorright)
+I’J+
N+N
9
N+N+N+N.
(2)
Hence the balance equations take the forms $(no V) = d$oo-
$(nV)
Vni( 2),,,cT)oo - Vnon( U,,,(T)Oth )
(3)
= 2Vn~(v,,,a)oo+2Vno~(u,,,a)~~h,
(4)
where no0 = 0.17 fmP3 is the normal nulcear density, no and n stand for the cold and thermalized densities, respectively, and ( Yb denotes an average over the distribution of species a and b. Since the compression is neglected, 2no+n=2noo.
(9
Eliminating no there remains one equation: (6) where V(t) is given by eq. (1). Eqs. (1) and (6) can be solved analytically if ( u,,,(T)oo= 2( u,,,u)oth .
(7)
Since v:i =
Oright -
Uleft =
2(
Uright -
uth)
=
zu$:
7
(8)
condition (7) is approximately valid if the cross section changes slowly between u and 2~. Assuming this, eq. (6) becomes substantially simpler, viz.
whence n(t) = 4?rU%,,~ ’ (Ro+m)
t2-27tf2~2(1-e8’)-~&t3
, 0
1
N.L. Balks
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Eq. (1) shows that the total overlap occurs at f = R,/ v. Then the density is n = %nax=;3Q&l+w)
{
1
l-2w+2w2(1-e-““)-3%
1
,
1
(11)
w=vr/RO.
At this moment we consider the ignition phase as finished; then the hot component is in thermal equilibrium, and the remaining cold nucleons leave the system. The homogeneous density dist~bution within the reaction volume corresponds to item (d).
2.2. EXPANSION
At the end of the ignition phase there are cold and hot nucleons. According to the present model, the cold ones move away with their initial velocities, while the motion of the hot nucleons generates an isentropic expansion of the spherical reaction volume. This expansion results in cooling, and if there is no evaporation, the equations of the problem have an analytic solution ‘). As we shall show, in our evaporation model the form of the solution remains unchanged, only some of the parameters acquire new values. In fact, n(r, t) =
t2
l-
(l+r:/OZ)=
%;(l+P/f;)
1’ a
(12)
u(r, t) =~(~/~~)(l+~z/~~)-l, T(r,t)=T,(l+P/t$l [
l-
p( r,t)=T&( [ 1+ t2,/ t;y2
l-
(13)
1 r2 1+CZ 1
r2 a;( 1+ t2/ tg> ’
a;( 1+ t2/ rf)
(14)
*
(1%
Here the time t is measured from the beginning of the expansion, cx is a parameter controlling the shape of the density distribution ((Y= 0 corresponds to a uniform density up to the sharp edge of the surface); the constants T,, no, to and Q. are determined by the initial conditions and are given by
(16) no = W~~Q&‘~(YO,
toA
C
J
mc2 2(a+l)To’
*, 2) ,
(17)
08)
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N.L. Baldzs et al. / Three-step analytic model
where Qo=Ro+h,
(19)
R,, being the initial radius and A being a depth parameter (having the same order of magnitude as the mean free path); I(y, a, 6) is an integral defined by
Yo=RolQo.
If there is no evaporation,
2.3. EVAPORATION
(20)
y. = 1.
STAGE
We consider now the boundary region of the matter in the reaction volume. Outside this region are all those particles which have already made their last collision and are now ready to escape without making any further collisions. Here the hydrodynamical description is no longer applicable. Inside this region is the internal domain. The boundary region separates these two domains. Our aim is to describe this situation as follows. Matter in the interior is described by hydrodynamics; from the surface outwards it is described as an expanding collisionless Knudsen gas. The transition region itself is represented by a fictitious mathematical surface, the skin or matching surface, whose location and motion are so determined that the conservation laws for matter, momentum and energy connecting the interior and exterior regions should be satisfied. It turns out that this is easy to do for the following reason. Conservation of matter determines the kinematics of the surface; the just-evaporated part of the gas is described by a half-maxwellian distribution in which two parameters, the temperature and the mean flow velocity, are still unknown. These two parameters are in turn determined by the energy and momentum conservation laws. The method is reminiscent of Gibbs introduction of a surface of separation. This is a mathematical surfa& which replaces the actual inhomogeneous density distribution between a liquid and its saturated vapour in such a way that the mechanical equilibrium conditions should be satisfied, and the matter should have constant but different densities on both sides of the surface. For Gibbs this surface is the carrier of the surface tension in order to satisfy the mechanical equilibrium conditions, and of the surface energy in order to satisfy the energy balance. In our case the surface tension will be replaced by an evaporation pressure, and the surface energy by an evaporation energy. The physical picture we have in mind is an expanding sphere whose outer layers are peeled off through an evaporation generated by particles escaping from this layer with a thermal velocity c(R).
N.L. Baldzs et al. / Three-step analytic model
535
If our picture were correct, i.e. the skin were infinitesimally thin, then the escaping thermal velocity c(R) could be calculated as
(21) whence, for a maxwellian distribution, c(R) = [2T(R)/7rm]“‘.
cm
However, this formula should not be taken too seriously because the skin is representative of a rather substantial volume from which evaporation is possible; thus in eq. (21) one should substitute a volume average of unknown form. For dimensional reasons this must result in an expression of the form c = k[2T/rm]“2,
(23)
where k is a number of order 1 (although in such a way we have slightly deviated from the chosen model). Consider now the surface at a radial distance R(t) from the centre. If without evaporation this surface were at rest, with evaporation R would shrink with the thermal velocity c(R). However there is also a flow with velocity u(R), hence we expect that the surface obeys the differential equation dR dt
-=-c(R)+u(R).
(24)
This intuitive result can formally be obtained as follows. At time t the particle number inside a sphere of radius R(t) is R(r) N(t)=4?T
I0
n(r, t)r2 dr.
(25)
Then the rate of change of this number can be obtained as joR$r2dr]
=4TR2n(E-u(R)),
(26)
using the continuity equation in the second term. On the other hand, this rate of change must be equal to -4nR2n(R)c(R), the number of escaping particles during unit time. This gives eq. (24). We now have characterized the evolution of the interior. If at time t we know the density, the velocity and the temperature inside a sphere of radius R, then eq. (24) determines the change of R, while the hydrodynamic equations govern the changes in the density, velocity and temperature. The initial value of R is a free parameter which is to be determined from the initial density, the total particle number, and the initial thickness of the surface layer.
N.L. Baldrs
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et al. / Three-step
analytic
model
The exterior is characterized by a momentum distribution in which the particle number is prescribed by number conservation, the mean flow is governed by the momentum conservation, and the external temperature parameter is determined by the energy conservation. Since we do not want to use explicitly the external density, only two balance equations remain, namely that the hydrodynamic pressure at the surface must be equal to the momentum carried away from unit surface in unit time, and that the rate of energy change is equal to the work done by the pressure pl, i.e. p1=
m(fi*-
%)%Cl,
(27) where fi2 and 2 stand for average values in the evaporating distribution. Note that eqs. (27) guarantee the continuity of the momentum and energy fluxes at the surface ‘*), thus they yield the correct jump conditions for the discontinuity.. Indices 1, 2 refer to quantities inside and outside, respectively. Expressing then O2and 3 one gets tJ2= u,(R)+-
&
3
2
T,(R) mcl(R) ’
TX(R) u:(R)+ WhW-&.
-+
m
1
The last term is connected with the fact that the outside momentum distribution is not spherical. In order to determine the parameter to, which gives the time scale of the expansion, one can use the integral form of energy conservation, 2mu1(r)2]n1(r)r2 dr
(29)
Using the analytic solution eqs. (12)-( 15) for the interior, the matching conditions (27) and eq. (26), this energy balance equation becomes an identity if (30) just as without evaporation ‘). The analogous integral form of the particle conservation turns out to be an identity independently of to. In this model the differential equation for the radius of the surface can also be analytically integrated. Namely, from eqs. (12)-(15) and (24) one gets (31)
N.L. Baldzs et al. / Three-step analytic model
537
where the particular form (23) has been used for c. The solution of this equation is R (7) = C&m
cos { 17arctg T+ arc cos (R,/ (2,)) , l/2
17= Wh,lQ,W~o/~m)
(32)
.
Since the difference A = Q,, - R,, determines the initial location of the surface of evaporation (and thus the speed of the disintegration), it is plausible to choose A to be of the order of the mean free path. Now, assuming that at t = 0 every particle belonged to the hydrodynamical region, we find that the initial data assume the values given by eqs. (16)-( 18); then the bulk properties of the expanding sphere are completely determined. Below the matching surface R(t) the evolution is purely hydrodynamical, so there the analytical solution (12)-(19) is valid. At the surface, eqs. (28) guarantee the correct energy, momentum and particle transfers. .Let us observe that the radius will decrease for large t-values if the skin is chosen to be wide enough, i.e. if the evaporation is faster than the expansion. On the other hand, the opposite situation holds for small values of A/R,,. The present description breaks down if the evolution reaches a stage in which the mean free path is comparable to R. If the evaporation rate is small, this will certainly occur for large times. We believe that this situation is irrelevant for heavy-ion collisions, and reflects a poor choice of the initial data. If R decreases with time, we expect a final stage R -A, but then the particle number inside the sphere will be sufficiently small. For these remaining particles we.apply the sudden break-up formalism ‘). 2.4. PARTICLE SPECTRUM
In the present model the particle spectrum is composed of the spectra of particles emerging during the evaporation stage. Let our detector accept only particles with a given velocity direction. The number of particles with velocity u in that direction emerging from the surface element dflnRR2 around R(t) is given by dNdanRR2 pz(u,R, t)---dt dt 47rR
(33)
’
where dN is the number of particles evaporated from the whole sphere in all directions. The detected spectrum will accordingly be given as F2(u) d3u=
(34)
For (p2(u, R(t), t) we choose a “half-maxwellian” 312
2 l++(u,JmlT,)
form e-(m~2T2)(U-U)Z~(~~~~~~)
,
(35)
538
N.L. Balks
et al. / Three-step analytic model
where if OS 8U0S&r if flUV>&r,
1,
I,
B(cos 4,“) = o
(36)
(37) is the error function; u2( t) is a velocity parameter; u,l(R, R pointing to the evaporation surface element; T,(t) is a temperature parameter. This distribution cp2 is normalized as cp2(u,R,t)d3U=1. The requiremqts averages v2 and v;,
of energ;
and momentum
2,cos z=
8Q2(
V2Q2(U,
V, t)
(38) conservation
demand
that the
d3u,
(39)
R, t) d3u,
be equal to the expressions given in eq. (31). These equations determine the parameters T2(t) and u2(t). The form of (p2 ensures the evaporation of all particles on the surface of the nucleus whose velocity is not pointing inwards. The explicit forms of c and 3 are given as e-wT2b:
~2=U2+J~m,2~2(1+~(u2Jm,~2)' ,-(m/2T*,4
1+
4(u2J75'
(40)
As we have seen, the surface evaporation cannot be complete, there will be some final stage when the remaining particles can depart from the whole volume. The contribution of these particles to the detected spectrum can be calculated as F;(v)
d3v= [&\OR
.(r)r2($)3’2T(r)3i2
1 e-(m/T(r))(o--U(r))Zdn dr] d3v.
(41)
The evaluation of this integral was treated in ref. ‘). 3. Results and discussion The present model contains three parameters
which are not directly determined by the initial beam data, viz. the shape parameter (Y,the depth of the initial surface
N.L. Baldzs
et al. / ‘flvee-step analytic model
539
h, and the factor k correcting the thermal escape velocity in eq. (23). The appropriate physical values of the first two parameters are connected with the prehistory of the expansion, while the third one should be calculated from the Boltzmann transport equation. Here we choose given values for a! and k, a = 0 (sharp-edged distribution),
t t
Ebo,,,b-8OOMeWnucl
1
0
1 bverlap
3
*
4
c
IO-”s Fig. 1. The radius of the hot sphere as a function of time for A+A collision at 800 MeV/nucleon beam energy. The arrows indicate the break-up (when n =0,4no,). The sphere can either expand or shrink, depending on the depth parameter A\,
I
0
2 t t6’ss
4
6
Fig. 2. The number of thermalized particles within the hot sphere as a function of time, for A+A collision at 800 MeV/nucleon bombarding energy. The sharp peak at the end of the ignition phase is an artefact of the model; the thermahzing interactions have been ignored after the ignition phase, and the evaporation starts only after the ignition phase.
540
N.L. Baldzs et al. / Three-step analytic model
and k = 1 (according to eq. (22)), and investigate the dependence of the observables on the depth parameter A. In fig. 1 the radius of the hot expanding sphere is plotted as a function of time for different A-parameters. It is important to note that for A-parameters of the order of magnitude of the mean free path the speed of evaporation is similar to or greater than the speed of expansion. Thus the sphere will shrink in spite of the strong radial flow. Fig. 2 shows the number of hot particles in the sphere. For technical reasons the calculation stopped at 0.4 normal nuclear density [before the break-up conditions of ref. ‘)I; however, for reasonable values of A, the number of particles has become by then sufficiently small in the sphere, thus in fact, we have a continuous break-up process. The energy spectra for Ebomb=800 MeVIA are given in fig. 3. For A = 3.0 fm the spectrum clearly shows the shouldering at low energies - as observed in ref. 13)and then the overall shape of the spectrum is in conformity with the experimental
A-3.0fm
l0” -
t :experimental
data
I 0
L
ZOO
400 600 E MeV
800
Fig. 3. Detected energy spectra for A+A collision at 800 MeV/nucleon beam energy. The dashed and continuous lines represent the calculated spectra, while the circles are experimental results i3). It can be seen that for A = 3 fm the overall shape of the spectrum conforms to the experimental data, although the apparent temperature is higher, because the treatment is non-relativistic and we neglected the pionic degrees of freedom. For A = 0 fm the shape does not agree with the experimental data.
N.L. Balks
et al. / Three-step
analytic
model
541
data. The apparent temperature is slightly higher (88 MeV) than the observed 76 MeV. Nevertheless, there are at least two reasons for getting too high a temperature. First, hadrochemical calculations show that the compression produces some pions as well. At this bombarding energy the number of pions is about 10% of the number of thermalized nucleons. The rest energy of these pions should then be subtracted from the total energy contained in the sphere which would mean some 7% decrease in the apparent temperature. Second, at 800 MeV/A bombarding energy both the flow and the thermal momentum distributions are quite relativistic, while we used a non-relativistic approximation. For A = 0 the computed spectrum has the wrong shape. Since this model contains continuous evaporation, the idealized notion of a sudden break-up used earlier in many models is now avoided. The u2( t) parameter obtained from the conservation laws is quite high, and this explains the result of ref. lo) where a large flow velocity was required to reproduce the experimental spectra. We may conclude as follows. In the present paper we have given an analytic model for the description of the complete nucleus-nucleus reaction: ignition, expansion and evaporation. The main advantage of this model seems to be that it describes in a simple and transparent manner the qualitive features of the processes which occur in a heavy-ion reaction; it was not its aim to give a quantitative description of the experimental data. Two of the authors N.B. and J.Z. would like to express their thanks to the Hungarian Academy of Sciences and to the National Science Foundation of the USA for their support. References 1) 2) 3) 4) 5) 6) 7) 8) 9) 10) 11) 12) 13)
I. Montvay and J. Zimanyi, Nucl. Phys. A316 (1979) 490 J.P. Bondorf, S.I.A. Garpman and J. Zimanyi, Nucl. Phys. A296 (1978) 320 A.A. Amsden, F.H. Harlow and J.R. Nix, Phys. Rev. Cl5 (1977) 2059 A.A. Amsden, A.S. Goldhaber, F.H. Harlow and J.R. Nix, Phys. Rev. Cl7 (1978) 2080 P. Danielewicz, Nucl. Phys. A314 (1979) 465 L.P. Csernai, B. LukLcs and J. Zimanyi, Lett. Nuovo Cim. 27 (1980) 111 K.K. Gudima, H. Iwe and V.D. Toneev, J. of Phys. G5 (1979) 229 K.M. Hartmann and B. Schiirmann, Z. Phys. A297 (1980) 311 J. Knoll, Nucl. Phys. A343 (1980) 511 P.J. Siemens and J.O. Rasmussen, Phys. Rev. Lett. 42 (1979) 880 H.W. Barz, L.P. Csemai and W. Greiner, Phys. Rev. C26 (1982) 740 L.D. Landau and E.M. Lifshitz, Fluid mechanics (Nauka, Moscow, 1953) S. Nagamiya, Proc. 4th High Energy Heavy Ion Summer Study, Berkeley, July 1978; private communication (1980)
THREE-STEP ANALYTIC MODEL FOR HIGH-ENERGY HEAVY-ION COLLISIONS N.L. BALAZS”,
J.P. BONDORF b, B. LUKACS’
International Workshop for Theoretical Physics, H-1525
and J. ZIMANYI’
Budapest 114, POB 49, Hungary
Received 25 July 1983 Abstract: We discuss within the framework of an analytical model central collisions between large nuclei at an intermediate energy. The model assumes three stages: the ignition stage in which thermalized nucleons are created within the reaction volume; the expansion phase in which the hot nuclear matter expands and cools; and finally, the evaporation stage in which the system disintegrates through a surface evaporation. The model provides a qualitative insight into the connections between the main physical processes throughout the collision event.
1. Introduction Models of increasing sophistication have been suggested for describing high-energy heavy-ion collisions. However, for such models [e.g. cascade, relativistic hydrodynamics, hadrochemistry, rows on rows, volume evaporation etc., see e.g. refs. ‘-97”)] a great deal of numerical work is required. On account of the complexity of these calculations it is difficult to gain an insight into the basic physics of the collisions. Similarly, simple and calculable models are desirable as inputs for QCD calculations in order to enable one to perform sensible estimations of phenomena connected with the formation of quark-gluon plasma. For these reasons we have developed a simple analytic model describing the whole evolution of the system in a central collision between two big nuclei. In the spirit of ref. ‘) the process is divided into three essentially different stages, viz. ignition, expansion and break-up; however, in the present model the idea of a continuous evaporation is also introduced. This model will then be applied to clarify some problems connected with the blast-wave model lo). In presenting our model we are particularly concerned with advanced dimensional analysis using simple but fundamental ideas of physics such as the conservation laws and basic kinetic theory. We hope that in doing so we provide a sufficiently adequate caricature of the actual features of the physical process. Our approach is basically Permanent addresses: “) Department of Physics, State University of New York at Stony Brook, Stony Brook, NY 11794, USA. ‘) The Niels Bohr Institute, Blegdamsvej 17, DK-2100 Copenhagen 0, Denmark. ‘) Central Research Institute for Physics, H-1525 Budapest 114, POB 49, Hungary. 530
N.L. Balks
et al. / Three-step analytic model
531
as follows: During the evolution of the system we distinguish three stages; these are (i) the ignition, (ii) the expansion, (iii) the break-up by evaporation. The description of stages (i) and (ii) follows traditional ideas; however, these are arranged in such a manner that the process can actually be described by simple expressions. In stage (iii) we depart from the standard approach and characterize the usual breakup stage differently. We do this by recognizing the important role of the mean free path in limiting the validity of the hydrodynamic description in an inhomogeneous and finite system. Because of this, we shall treat stage (i) in a rather schematic fashion, its whole purpose being to provide initial data for stages (ii) and (iii) in which the role of the mean free path and the idea of a continuous evaporation will be displayed. The latter has, in particular, the important advantage that it replaces the rather ad hoc introduction of a sudden break-up by the more continuous process of evaporation, eliminating thereby the sharp division between stages (ii) and (iii).
2. Description 2.1. IGNITION
of the model
STAGE
Initially the system consists of two equal nuclei having sharp edges and being in contact; after this, the ignition stage starts; it ends when the two nuclei overlap completely. During this stage the velocity ‘distribution is becoming thermalized. Even though the detailed mechanism of this process may be quite complicated, we shall construct a simple and analytical model. As in ref. ‘) we do this by using some more or less technical approximations: (a) The compression is neglected. (b) Only nucleons are included, other particles such as r, A, etc. are not treated explicitly. This simplification is justified because, according to the hadron chemistry calculation of ref. ‘), the relative density of A’s never exceeds 10% in the considered energy range. (c) The particles are considered thermalized after the first collision. (d) Within the reaction volume (i.e. in the overlap region) density gradients are neglected. Although these simplifications might be considered substantial, it is easy to see that they are necessary: for example, (c) is required by our wish that the further phases of the evolution should be calculable assuming thermal equilibrium. Using item (a) one obtains for the reaction volume
where V, = $rRi is the nulcear volume before the collision, and v is the velocity of the nuclei in the c.m. system, v =$vlab.
532
N.L. Baldzs et al. / Three-step analytic model
Items (b) and (c) mean that we distinguish only three different components of which two are cold, and one is thermalized. Because of the right-left symmetry only two different densities occur in the balance equations. The assumed reactions are as follows: N o.left+ No,right+ N + N 9 N O,(leftorright)
+I’J+
N+N
9
N+N+N+N.
(2)
Hence the balance equations take the forms $(no V) = d$oo-
$(nV)
Vni( 2),,,cT)oo - Vnon( U,,,(T)Oth )
(3)
= 2Vn~(v,,,a)oo+2Vno~(u,,,a)~~h,
(4)
where no0 = 0.17 fmP3 is the normal nulcear density, no and n stand for the cold and thermalized densities, respectively, and ( Yb denotes an average over the distribution of species a and b. Since the compression is neglected, 2no+n=2noo.
(9
Eliminating no there remains one equation: (6) where V(t) is given by eq. (1). Eqs. (1) and (6) can be solved analytically if ( u,,,(T)oo= 2( u,,,u)oth .
(7)
Since v:i =
Oright -
Uleft =
2(
Uright -
uth)
=
zu$:
7
(8)
condition (7) is approximately valid if the cross section changes slowly between u and 2~. Assuming this, eq. (6) becomes substantially simpler, viz.
whence n(t) = 4?rU%,,~ ’ (Ro+m)
t2-27tf2~2(1-e8’)-~&t3
, 0
1
N.L. Balks
et al. / Three-step analytic model
533
Eq. (1) shows that the total overlap occurs at f = R,/ v. Then the density is n = %nax=;3Q&l+w)
{
1
l-2w+2w2(1-e-““)-3%
1
,
1
(11)
w=vr/RO.
At this moment we consider the ignition phase as finished; then the hot component is in thermal equilibrium, and the remaining cold nucleons leave the system. The homogeneous density dist~bution within the reaction volume corresponds to item (d).
2.2. EXPANSION
At the end of the ignition phase there are cold and hot nucleons. According to the present model, the cold ones move away with their initial velocities, while the motion of the hot nucleons generates an isentropic expansion of the spherical reaction volume. This expansion results in cooling, and if there is no evaporation, the equations of the problem have an analytic solution ‘). As we shall show, in our evaporation model the form of the solution remains unchanged, only some of the parameters acquire new values. In fact, n(r, t) =
t2
l-
(l+r:/OZ)=
%;(l+P/f;)
1’ a
(12)
u(r, t) =~(~/~~)(l+~z/~~)-l, T(r,t)=T,(l+P/t$l [
l-
p( r,t)=T&( [ 1+ t2,/ t;y2
l-
(13)
1 r2 1+CZ 1
r2 a;( 1+ t2/ tg> ’
a;( 1+ t2/ rf)
(14)
*
(1%
Here the time t is measured from the beginning of the expansion, cx is a parameter controlling the shape of the density distribution ((Y= 0 corresponds to a uniform density up to the sharp edge of the surface); the constants T,, no, to and Q. are determined by the initial conditions and are given by
(16) no = W~~Q&‘~(YO,
toA
C
J
mc2 2(a+l)To’
*, 2) ,
(17)
08)
534
N.L. Baldzs et al. / Three-step analytic model
where Qo=Ro+h,
(19)
R,, being the initial radius and A being a depth parameter (having the same order of magnitude as the mean free path); I(y, a, 6) is an integral defined by
Yo=RolQo.
If there is no evaporation,
2.3. EVAPORATION
(20)
y. = 1.
STAGE
We consider now the boundary region of the matter in the reaction volume. Outside this region are all those particles which have already made their last collision and are now ready to escape without making any further collisions. Here the hydrodynamical description is no longer applicable. Inside this region is the internal domain. The boundary region separates these two domains. Our aim is to describe this situation as follows. Matter in the interior is described by hydrodynamics; from the surface outwards it is described as an expanding collisionless Knudsen gas. The transition region itself is represented by a fictitious mathematical surface, the skin or matching surface, whose location and motion are so determined that the conservation laws for matter, momentum and energy connecting the interior and exterior regions should be satisfied. It turns out that this is easy to do for the following reason. Conservation of matter determines the kinematics of the surface; the just-evaporated part of the gas is described by a half-maxwellian distribution in which two parameters, the temperature and the mean flow velocity, are still unknown. These two parameters are in turn determined by the energy and momentum conservation laws. The method is reminiscent of Gibbs introduction of a surface of separation. This is a mathematical surfa& which replaces the actual inhomogeneous density distribution between a liquid and its saturated vapour in such a way that the mechanical equilibrium conditions should be satisfied, and the matter should have constant but different densities on both sides of the surface. For Gibbs this surface is the carrier of the surface tension in order to satisfy the mechanical equilibrium conditions, and of the surface energy in order to satisfy the energy balance. In our case the surface tension will be replaced by an evaporation pressure, and the surface energy by an evaporation energy. The physical picture we have in mind is an expanding sphere whose outer layers are peeled off through an evaporation generated by particles escaping from this layer with a thermal velocity c(R).
N.L. Baldzs et al. / Three-step analytic model
535
If our picture were correct, i.e. the skin were infinitesimally thin, then the escaping thermal velocity c(R) could be calculated as
(21) whence, for a maxwellian distribution, c(R) = [2T(R)/7rm]“‘.
cm
However, this formula should not be taken too seriously because the skin is representative of a rather substantial volume from which evaporation is possible; thus in eq. (21) one should substitute a volume average of unknown form. For dimensional reasons this must result in an expression of the form c = k[2T/rm]“2,
(23)
where k is a number of order 1 (although in such a way we have slightly deviated from the chosen model). Consider now the surface at a radial distance R(t) from the centre. If without evaporation this surface were at rest, with evaporation R would shrink with the thermal velocity c(R). However there is also a flow with velocity u(R), hence we expect that the surface obeys the differential equation dR dt
-=-c(R)+u(R).
(24)
This intuitive result can formally be obtained as follows. At time t the particle number inside a sphere of radius R(t) is R(r) N(t)=4?T
I0
n(r, t)r2 dr.
(25)
Then the rate of change of this number can be obtained as joR$r2dr]
=4TR2n(E-u(R)),
(26)
using the continuity equation in the second term. On the other hand, this rate of change must be equal to -4nR2n(R)c(R), the number of escaping particles during unit time. This gives eq. (24). We now have characterized the evolution of the interior. If at time t we know the density, the velocity and the temperature inside a sphere of radius R, then eq. (24) determines the change of R, while the hydrodynamic equations govern the changes in the density, velocity and temperature. The initial value of R is a free parameter which is to be determined from the initial density, the total particle number, and the initial thickness of the surface layer.
N.L. Baldrs
536
et al. / Three-step
analytic
model
The exterior is characterized by a momentum distribution in which the particle number is prescribed by number conservation, the mean flow is governed by the momentum conservation, and the external temperature parameter is determined by the energy conservation. Since we do not want to use explicitly the external density, only two balance equations remain, namely that the hydrodynamic pressure at the surface must be equal to the momentum carried away from unit surface in unit time, and that the rate of energy change is equal to the work done by the pressure pl, i.e. p1=
m(fi*-
%)%Cl,
(27) where fi2 and 2 stand for average values in the evaporating distribution. Note that eqs. (27) guarantee the continuity of the momentum and energy fluxes at the surface ‘*), thus they yield the correct jump conditions for the discontinuity.. Indices 1, 2 refer to quantities inside and outside, respectively. Expressing then O2and 3 one gets tJ2= u,(R)+-
&
3
2
T,(R) mcl(R) ’
TX(R) u:(R)+ WhW-&.
-+
m
1
The last term is connected with the fact that the outside momentum distribution is not spherical. In order to determine the parameter to, which gives the time scale of the expansion, one can use the integral form of energy conservation, 2mu1(r)2]n1(r)r2 dr
(29)
Using the analytic solution eqs. (12)-( 15) for the interior, the matching conditions (27) and eq. (26), this energy balance equation becomes an identity if (30) just as without evaporation ‘). The analogous integral form of the particle conservation turns out to be an identity independently of to. In this model the differential equation for the radius of the surface can also be analytically integrated. Namely, from eqs. (12)-(15) and (24) one gets (31)
N.L. Baldzs et al. / Three-step analytic model
537
where the particular form (23) has been used for c. The solution of this equation is R (7) = C&m
cos { 17arctg T+ arc cos (R,/ (2,)) , l/2
17= Wh,lQ,W~o/~m)
(32)
.
Since the difference A = Q,, - R,, determines the initial location of the surface of evaporation (and thus the speed of the disintegration), it is plausible to choose A to be of the order of the mean free path. Now, assuming that at t = 0 every particle belonged to the hydrodynamical region, we find that the initial data assume the values given by eqs. (16)-( 18); then the bulk properties of the expanding sphere are completely determined. Below the matching surface R(t) the evolution is purely hydrodynamical, so there the analytical solution (12)-(19) is valid. At the surface, eqs. (28) guarantee the correct energy, momentum and particle transfers. .Let us observe that the radius will decrease for large t-values if the skin is chosen to be wide enough, i.e. if the evaporation is faster than the expansion. On the other hand, the opposite situation holds for small values of A/R,,. The present description breaks down if the evolution reaches a stage in which the mean free path is comparable to R. If the evaporation rate is small, this will certainly occur for large times. We believe that this situation is irrelevant for heavy-ion collisions, and reflects a poor choice of the initial data. If R decreases with time, we expect a final stage R -A, but then the particle number inside the sphere will be sufficiently small. For these remaining particles we.apply the sudden break-up formalism ‘). 2.4. PARTICLE SPECTRUM
In the present model the particle spectrum is composed of the spectra of particles emerging during the evaporation stage. Let our detector accept only particles with a given velocity direction. The number of particles with velocity u in that direction emerging from the surface element dflnRR2 around R(t) is given by dNdanRR2 pz(u,R, t)---dt dt 47rR
(33)
’
where dN is the number of particles evaporated from the whole sphere in all directions. The detected spectrum will accordingly be given as F2(u) d3u=
(34)
For (p2(u, R(t), t) we choose a “half-maxwellian” 312
2 l++(u,JmlT,)
form e-(m~2T2)(U-U)Z~(~~~~~~)
,
(35)
538
N.L. Balks
et al. / Three-step analytic model
where if OS 8U0S&r if flUV>&r,
1,
I,
B(cos 4,“) = o
(36)
(37) is the error function; u2( t) is a velocity parameter; u,l(R, R pointing to the evaporation surface element; T,(t) is a temperature parameter. This distribution cp2 is normalized as cp2(u,R,t)d3U=1. The requiremqts averages v2 and v;,
of energ;
and momentum
2,cos z=
8Q2(
V2Q2(U,
V, t)
(38) conservation
demand
that the
d3u,
(39)
R, t) d3u,
be equal to the expressions given in eq. (31). These equations determine the parameters T2(t) and u2(t). The form of (p2 ensures the evaporation of all particles on the surface of the nucleus whose velocity is not pointing inwards. The explicit forms of c and 3 are given as e-wT2b:
~2=U2+J~m,2~2(1+~(u2Jm,~2)' ,-(m/2T*,4
1+
4(u2J75'
(40)
As we have seen, the surface evaporation cannot be complete, there will be some final stage when the remaining particles can depart from the whole volume. The contribution of these particles to the detected spectrum can be calculated as F;(v)
d3v= [&\OR
.(r)r2($)3’2T(r)3i2
1 e-(m/T(r))(o--U(r))Zdn dr] d3v.
(41)
The evaluation of this integral was treated in ref. ‘). 3. Results and discussion The present model contains three parameters
which are not directly determined by the initial beam data, viz. the shape parameter (Y,the depth of the initial surface
N.L. Baldzs
et al. / ‘flvee-step analytic model
539
h, and the factor k correcting the thermal escape velocity in eq. (23). The appropriate physical values of the first two parameters are connected with the prehistory of the expansion, while the third one should be calculated from the Boltzmann transport equation. Here we choose given values for a! and k, a = 0 (sharp-edged distribution),
t t
Ebo,,,b-8OOMeWnucl
1
0
1 bverlap
3
*
4
c
IO-”s Fig. 1. The radius of the hot sphere as a function of time for A+A collision at 800 MeV/nucleon beam energy. The arrows indicate the break-up (when n =0,4no,). The sphere can either expand or shrink, depending on the depth parameter A\,
I
0
2 t t6’ss
4
6
Fig. 2. The number of thermalized particles within the hot sphere as a function of time, for A+A collision at 800 MeV/nucleon bombarding energy. The sharp peak at the end of the ignition phase is an artefact of the model; the thermahzing interactions have been ignored after the ignition phase, and the evaporation starts only after the ignition phase.
540
N.L. Baldzs et al. / Three-step analytic model
and k = 1 (according to eq. (22)), and investigate the dependence of the observables on the depth parameter A. In fig. 1 the radius of the hot expanding sphere is plotted as a function of time for different A-parameters. It is important to note that for A-parameters of the order of magnitude of the mean free path the speed of evaporation is similar to or greater than the speed of expansion. Thus the sphere will shrink in spite of the strong radial flow. Fig. 2 shows the number of hot particles in the sphere. For technical reasons the calculation stopped at 0.4 normal nuclear density [before the break-up conditions of ref. ‘)I; however, for reasonable values of A, the number of particles has become by then sufficiently small in the sphere, thus in fact, we have a continuous break-up process. The energy spectra for Ebomb=800 MeVIA are given in fig. 3. For A = 3.0 fm the spectrum clearly shows the shouldering at low energies - as observed in ref. 13)and then the overall shape of the spectrum is in conformity with the experimental
A-3.0fm
l0” -
t :experimental
data
I 0
L
ZOO
400 600 E MeV
800
Fig. 3. Detected energy spectra for A+A collision at 800 MeV/nucleon beam energy. The dashed and continuous lines represent the calculated spectra, while the circles are experimental results i3). It can be seen that for A = 3 fm the overall shape of the spectrum conforms to the experimental data, although the apparent temperature is higher, because the treatment is non-relativistic and we neglected the pionic degrees of freedom. For A = 0 fm the shape does not agree with the experimental data.
N.L. Balks
et al. / Three-step
analytic
model
541
data. The apparent temperature is slightly higher (88 MeV) than the observed 76 MeV. Nevertheless, there are at least two reasons for getting too high a temperature. First, hadrochemical calculations show that the compression produces some pions as well. At this bombarding energy the number of pions is about 10% of the number of thermalized nucleons. The rest energy of these pions should then be subtracted from the total energy contained in the sphere which would mean some 7% decrease in the apparent temperature. Second, at 800 MeV/A bombarding energy both the flow and the thermal momentum distributions are quite relativistic, while we used a non-relativistic approximation. For A = 0 the computed spectrum has the wrong shape. Since this model contains continuous evaporation, the idealized notion of a sudden break-up used earlier in many models is now avoided. The u2( t) parameter obtained from the conservation laws is quite high, and this explains the result of ref. lo) where a large flow velocity was required to reproduce the experimental spectra. We may conclude as follows. In the present paper we have given an analytic model for the description of the complete nucleus-nucleus reaction: ignition, expansion and evaporation. The main advantage of this model seems to be that it describes in a simple and transparent manner the qualitive features of the processes which occur in a heavy-ion reaction; it was not its aim to give a quantitative description of the experimental data. Two of the authors N.B. and J.Z. would like to express their thanks to the Hungarian Academy of Sciences and to the National Science Foundation of the USA for their support. References 1) 2) 3) 4) 5) 6) 7) 8) 9) 10) 11) 12) 13)
I. Montvay and J. Zimanyi, Nucl. Phys. A316 (1979) 490 J.P. Bondorf, S.I.A. Garpman and J. Zimanyi, Nucl. Phys. A296 (1978) 320 A.A. Amsden, F.H. Harlow and J.R. Nix, Phys. Rev. Cl5 (1977) 2059 A.A. Amsden, A.S. Goldhaber, F.H. Harlow and J.R. Nix, Phys. Rev. Cl7 (1978) 2080 P. Danielewicz, Nucl. Phys. A314 (1979) 465 L.P. Csernai, B. LukLcs and J. Zimanyi, Lett. Nuovo Cim. 27 (1980) 111 K.K. Gudima, H. Iwe and V.D. Toneev, J. of Phys. G5 (1979) 229 K.M. Hartmann and B. Schiirmann, Z. Phys. A297 (1980) 311 J. Knoll, Nucl. Phys. A343 (1980) 511 P.J. Siemens and J.O. Rasmussen, Phys. Rev. Lett. 42 (1979) 880 H.W. Barz, L.P. Csemai and W. Greiner, Phys. Rev. C26 (1982) 740 L.D. Landau and E.M. Lifshitz, Fluid mechanics (Nauka, Moscow, 1953) S. Nagamiya, Proc. 4th High Energy Heavy Ion Summer Study, Berkeley, July 1978; private communication (1980)