A simple analytic hydrodynamic model for expanding fireballs

Nuclear Physics A296 (1978) 320-332 ; © North-l~oi7and Prbllihlttp Co., Matddmsr Na to be reproduced by photopsint or mbro8lm wlthoat wrlttm permir~tom thom the publisher

A SIMPLE ANALYTIC HYDRODYNAMIC MODEL FOR EXPANDING FIREBALLS J . P . BONDORF The Niets Bohr Institute, University ojCopenhagen, Blegdamsvef 17, 2100 Copenhagrn ~ S . I. A. GARPMAN NORDTd, BlepdarnsueJ 17, 2100 Copenhagrn ~ and J. ZIM~4NY1 Crntral Research Institute jar Physics, H-1315 Budapest 114, PO Box 49, Hungary Received 4 Aptil 1971 (Revised 29 August 19'77) A6etract : A simple analytic solution is presented for the twn-linear hydrodynamic oquatioas describing a free isentropic expansion into vacuum . A "break-up" oonoept is introduced yielding a welldefmed break-up time, density, temperature and final state inclusive particle spectrum. 1 . Introduction

There are experimental indications that in an energetic heavy-ion collision there is formed a hot nuclear gas cloud from all or a part of the nuclear matter, which then later on explodes t ) . For the description of this phenomenon, several models have been applied. In attempting to fit composite-particle spectra to moving Boltzmann distributions, some fundamental discrepancies with these theories were encountered, namely the fitted param~ers (temperature and c.m. velocity) were not understandable, either in the fireball picture or in the target explosion picture. This suggests collective hydrodynamic features of the collisions : The first hydrodynamic model was proposed by Chapline et al. Z). Other such models for the collisions have been discussed in refs. s s) . Refs. Z " a, s) mainly concentrate on the compression stages in the reactions, but do not discuss in detail the expansion of the matter. In ref. `), a numerical simulation of both the compression and expansion stages is performed. Another approach to the solution of the probltm of disintegration of a piece of hot nuclear matter into flee space can be found in ref. 6). The present work concentrates on the dynamics of the explosion phase. In this phase the nuclear matter is at high temperature and relatively low density. Under such conditions it is hopefully possible to find a reliable equation of state. 320

FIREBALLS

32 1

We will assume that at some instant during the collision there exists a zone of hot compressed nuclear matter at relative rest, the whole thing moving with its c.m. velocity. This is conceptually close to, but not identical with, the fireball introduced in ref.'), The difference will be clear from the discussion given later. A hydrodynamic description is used. The model is non-relativistic and applicable to beam energies up to about 400 MeV per nucleon. It is desirable to investigate such a model because even the most pessimistic approach as seen from a hydra dynamic point of view - the Monte Carlo calculation with only nuclear degrees of freedom - suggests hydrôdynamic features `~ the decaying phase of the hot matter. In the following sections a simple analytic solution to the hydrodynamic equations is presented. In addition, we introduce the concept of a break-up time, which is the time at which the constituents of the gas stop interacting. With these two tools together, one can easily calculate experimentally observable quantities. It is, furthermore, demonstrated that the analytic solutions give deeper insight into the time developments and spatial distributions of the exploding hot zones . They could therefore also give insight into existing numerical calculations with hydrodynamic collision models. The improvement of the Berkeley experiments to include heavier projectiles (like Fe) and the development of new experimental methods of selecting central collisions call for hydrodynamic models which are more adequate for big systems. 2. T7~e sôlotioos of the hydrodynamk eq~wtioos The initial conditions for the hot zone of nuclear matter formed just prior to the expansion are not known at present. Inspection of earlier estimates and models, however, suggests that it would be worthwhile assuming, as a first attempt, that the hot zone is approximately spherical, at relative rest and in local thermal equilibrium. A hot compressed spherical ball is therefore our starting point. It is furthermore assumed that the equation of state of the gas is that of the ideal gas'). Now consider the explosion of a spherically symmetric ball into vacuum. The equation of motion, the continuity equation and the equation of state for the isentropic expansion read as follows ')

2u~ p~+p,u+p u,+ - = 0,

(lb) (lc)

Here u(r, t), p(r, t) and p(r, t) are the local macroscopic flow velocity, density and pressure fields, respectively . The subscripts t and r refer to partial derivatives with

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J. P. BONDORF et al.

respect to time and space. The constant y is the ratio of the spec heats at constant pressure and volume, respectively . To obtain a class of solutions of polynomial form, the following ansatz was examined P(r, t) _ ~a R3(t) Cl - CR(t)) ~ 0

~r 5 R ifr>R,

(2a)

where R(t) is a function of time only, and a and b are constants (their interpretation for the nuclear system will be given later) . For the constants u, a and n, restrictions will be set by eqs. (la}{c). Eq. (2b) expresses that the system behaves adiabatically in every space-time point and the relation is valid for any medium with isothermal compressibility, being only inversely proportional to the pressure . It includes as extremum cases the ideal quantum gas and the ideal classical gas with constant heat capacity . We want to consider the expansion stage where the temperature starts around 5t~100 MeV [ref. 1)] . In this energy interval where the free NN cross section approaches its minimum value, the best possible conditions for .using ideal gas laws are fulfilled. Since the equation ofstate ofnuclear matter at high density and temperature is rather unknown, we have adopted an ideal gas law in the whole density range, although it is open to question in the very early stages of the expansion. It is known from relativistic mean-field theories 1~ that for temperatures considerably more than zero but smaller than the rest energy of the baryon (938 MeV) and vanishing density; the equation of state is that of a classical non-relativistic gas. Thus, it may well be appropriate during the stage of the expansion where the velocity distributions become frozen . A straightforward substitution of expressions (2a) and (2b) into eqs. (la}-(lc) yields the result n = 2,

(3a)

P = 1-a(y-1~

(3b)

tdr, t) = r R , )~

(~)

From now on we will treat the case of a monatomic ideal classical gas with constant heat capacities (y = co/co = ~). It is interesting to note that eq. (3d) can be used even

FIREBALLS

32 3

in the quantum limit, but then the constant y is not necessarily the ratio ofthe specific heats. Since the thermal wavelength is smaller than or ofthe order of the interpardcle spacing, it is not believed that quantum effects will be predominant for the expansion phase until clustering eventually occurs . In principle, eq. (3d) will have analytical solutions only for some values of y(3, 3, etc.), but it can, of course, be solved numerically for other values of y. For the case y = 3, eq . (3d) was integrated, yielding where d

R(t) = d(t~ + tû)~,

(4a)

= C2a~b(a+1) ~~,

(4b)

0

and to are constants to be determined later. In order to check the conservation ofenergy, the total heat and the kinetic energies associated with the mass flow were calculated Rnu~ = 3nbasr3 9(~, a+2) ds(tz+to) ,

(Sa)

~+~ = .a~~, a+lkh ts+t2 . 0

(Sb)

Here B(z, w) is the ß-function (see appendix B). As time grows to infinity, the heat energy goes to zero, while thekinetic energy grows to a constant value. One can easily check by adding eqs. (Sa) and (Sb) that energy conservation is fulfilled for all times. The solution (2), (3) is a sculled similarity solution of non-linear equations'). To our knowledge this solution has not been presented before . 3. Geometrical concept of the break-op For finite-sized gas balls with constituents having short-ranged interaction it is intuitively clear that the hydrodynamic picture cannot hold for infinitely long times. As the expansion of the gas ball into the vacuum proceeds, the density of gas, and with it the number of collisions, decreases rapidly leading to a "freezing in" of the energy distribution of the emerging particles . A mathematical expression for this idea will be given as a break-up ("freezing-in") criterium. Let us consider two particles (fig . 1) with mean interparticle distance l lying on a sphere of radius r. On average these particles will have a radially outward directed velocity, u(r, t), due to the macroscopic flow. Due to this radial flow, the particles will fly apart from each other with a velocity, w, w(r, t)

=

Kr' t) r idr, t),

(6a)

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J. P . BONDORF et al.

Fig. 1 . The flying-apart velocity, w, caused by the radial flow G, is visualized for two particles situated ' at a distance r from the explosion cents. Here 1 is the mean intaparticle distance.

On the other hand, these particles will have on the average a randomly directed velocity with mean value v(r, t). The deooupling of the thermal degrees of freedom from the macroscopic flow will occur when w(r, t) exceeds v(r, t). The break-up time, tb , is defined as the time when w for a given space point r equals v w(r, t~ = v(r, t~.

(6b)

In order that the system actually obeys the hydrodynamic laws, it must fulfil also the usual criterion that the hydrodynamic rarefaction time °) for a macroscopic volume element (i.e. containing many more than one, but considerably fewer than the total numberofnucleons) is larger thanthe microscopic interaction time which is a function of local density, mean spced and NN cross section. This condition is fulfilled, in particular due to the increase of the NN scattering cross section with decreasing temperature. We assume that the free NN cross section can be used also in diluted nuclear matter . By the use of eq. (6b), the thermodynamic variables at break-up were calculated. The formulae so obtained are given in appendix A. 4. Doable difïereatirl cross sections Consider a small fluid element moving at a macroscopic speed u with a density p and temperature T (see fig. 2). The distribution for the energy i of the particles in the fluid element at rest ~is given by a Boltzmann distribution (normalized to unity) (7a) Galilean transformation of this distribution is made by the Jacobian â(s, ~) s (~b) ; ~(~ ~) ~' where s is the particle energy in the new coordinate system. The energy distribution

FIREBALLS

325

Fig. 2 . In this figure the difïerent velocities entering in the Galilean transformations needed for the calculation of cross sections .are displayed . Here o o is the velocity of the c:m . of the exploding system, J is the velocity of the c.m . of a fluid eltment and ü is the thermal velocity of a particle within it. (B is the angle between ü and û.)

reaching a detector coming from a fluid element where particle velocities make an _ angle (B, ~) with respect to the macroscopic flow velocity u is (sce flg. 2) : (7c) here W denotes the flow energy (~mu2) per particle. Since B and ~ are random, an integration is performed to get the total energy distribution for the particles reaching the detector which originate in the fluid element =

1

nkT

1 e-a+w~n~r sinh ~ ( kT )

(

JW

Nowwe want to integrate over all fluid elements within which the frozen distributions are realized (i.e. we want an integration over r at break-up). Thus xc°b~ 1 1 _ 1 mN Jo nkT(r, t,a W(r x sinh ~ changing to the variable x = rlR(t~, one gets .f(8) =

mN J 0

kTb

~N'b

sW(r, t~ p(r, t~dr; kT(r, t,~

e-ts+tl~/tTb sinh ~

kTb

b

PbRb~~

(9a)

(9b)

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J. P. BONDORF et al.

The quantities Th, wh , pb, Rh are given in appendix A as functions of z. Here N is the number of particles in the system . Now it is trivial to get the double dif%rential cross section. Assuming a geometrical partial cross section v° (discussed in the next section) one gets d2 Q _ Nf(e). dadrl - ~°

(l0a)

This is the expected cross sections in the c.m. frame of the expanding sphere . If the spheres move as a whole with an associated energy per particle of e°, the Jacobian (7b) can be used to obtain Q°N~e dzQ here

dEau =

~s

I(~ ;

(lOb)

This, is the distribution, which can be compared directly with experimental data. 5. Applicadon of the model to heavy-ion reacdoos The present model gives the description of the time evolution of a hot, compressed, nucleon gas ball . However, to obtain results comparable with experiment, a simple geometric picture is used to describe its formation. In the collision of the projectile with the target nucleus, a part from each of them (np and ~, respectively) will enter as a constituent of the hot gas. Neither nP. norn, nced to be the total number ofnucleons contained in the projectile and target nucleus. In fact, nr and n, are determined by the impact parameter D, which experimentally is not easily controllable. However, we shall proceed to obtain relationships not depending on D . Thus, let us denote by e,,b the energy per baryon initially available. Then it will be assumed that E _ npe,,b is the total energy available for the sum of the c.m . kinetic energy, E°, and the heat energy, Eneu, of the ball . Furthermore, in the c.m . system of the ball at time t = 0 it is assumed that all the energy present is in the form of thermal motion of the np +n, baryons. From these considerations one obtains : Eh~u(0) _-- (n,, + ~kh~(0) =

e~h

np

~~ +

(1 l a)

Denoting by v the dimensionless ratio of the central density at time t = 0, p(0, 0), to that of the normal nuclear matter mass density, p°, then, by integration of eq . (2a) over the whole space, and by the use of eq . (Sa), one finally gets the following expres-

FIitEBALIS

327

sions for the constants : (nr+n~m

a __

__

(12a}

2aB(~, a+ 1)'

(nv + t~m

(12c) C(a+ ~ti«~(0)~C2~Po~~~ a+1)] In the present model the only free parameter besides the maximum compression ratio v is the shape parameter a. In fact, as will be shown, for the nucleon cross sections only . the shape parameter remains as a free parameter. Fig. 2 displays the density as a function of x = r/R, for dit%rent a-values . It is wothwhile to notice that the thermodynamic variables p, p and T all depend on x through a form factor (1-x2)" (fig . 3a), the exponent n in general being dif%rent for the three functions in a given case . In fig. 3 p and p are shown for a = 0.5, I .5 and 3. As can be seen from to

2

z 0

~w

e

0

N

X

w W

O

N w n_

RATIO

p.s

x -- ~`

~

Fig. 3a. The form factor as a function of the dimensionless variable x plotted for different a-values .

ô 0 .5 ...

0

0

.5 0

PARAMETER

x = r/R

1,0

Fig. 3b . The density, p, and pressure, p, es functions of x = r/R for different shape parameters, a .

328

~

J. P. HONDORF et d.

t,

2t,

t Fig . 4. The compression of the nucleon gas at the center of the exploding system as a function of time . The upper curve is the result of a Monte Carlo cakvlation reproduced from rd. °), while the lower curve is the fnndional form of the prediction of the hydrodynamic model (e4 . (2a)) . TIME

fig. 2, for the density distribution the a -~ 0 case approaches the step function B(x), while the a -~ ao case approaches a Disc delta function, S(x) . The physical value of a may depend on the initial conditions and is not known at present. However, inspection of the results of the Monte Carlo calculations suggests an a-value of the order of one [see fig. .3 of ref. a)] . Also, the time behaviour of the central density p(0, t) (eq. (2a)) can be compared with that obtained in the Monte Carlo calculation. In fig. 4 the central compression ratio is given as a function of time, obtained from eq. (2a) and from the Monte Carlo calculation of ref. s). The displacement of the zero points of the time scales for the two curves indicates at which time during a central heavy-ion collision the expansion starts . A calculation of to (eq. (12c)) for U+U at e,,b = 400 MeV/nucleon gives for a = 1 the value to = 3.8 x 10 - zs sec. This value changes by only a few per cent when a varies from 0.5 to 2.5. A fit of the functional shape for p(0, t) (eq. (2a)) to the Monte Carlo calculation as shown in fig. 3 gives to = 6 x 10 -zs sec. The two values of to, are rather close considering the big difference between the physical assumptions and boundary conditions in the two cases. However, comparison of classical hard-sphere calculations with hydrodynamic calculations for the case of Z°Ne+ 23eU collisions ~ shows significant differences in features such as the nucleon spectra. This may be due to the big difference between the effective equations of state (in case of approximate local equilibrium) and the effect of different transparencies in the two cases. Finally, a posteriors, the correctness of neglecting transport properties in the gas

329

FIREBALLS

cloud was examined. One can easily realize that, due to the spherical symmetry and the special initial conditions applied, the viscosity term in the equation of motion is identically equal to zero . The effect of heat conduction in an energy balance should, however, in principle be included . We have estimated its magnitude assuming a heat conductivity term to be KVZ T, where the thermal conductivity K was chosen to be IOZZ s-t - fm-t . It was found that, except for t ~ t o and r very close to R, the term was much less than 10 ~~ of the other terms in the energy balance and therefore neglected. The hydrodynamic nature of the model discussed in this paper makes it most suitable for central or almost-central collisions of large projectiles and targets of not very different size. Therefore it is not possible to find experimental data at present for comparison . In the following, calculated proton spectra and angular distributions for some central collisions of heavy systems will be given. For the calculations the idea of target explosion was invoked. This means that all the nucleons are involved in forming the hot gas. This of course means that the collisions are near central. I CALCULATED CROSS SECTION FOR NEAR CENTRAL Fa " Ag AT E~pe' S00MaV/ NUCLEON o: = 1 .113 b

z o_ w a

iooo aL = 0.5

a_

U z

w o: w

elp9 ' 30°

ioo

W W Ô W

m S

0

ioo

xoo

aoo

ENERGY OF REACTION PRODUCTS MeV / NUCLEON Fig . S . Double differential nucleon cross sections predicted by the present model for the Fe+Ag reaction . The common parameters used in the calculations were e,, b = 500 MeV and a = 1113 mb . It was assumed that all of the nucleons of both the target and projedik were involved in the explosion .

33 0

J. P. BONDORF et at. I

I

I

CALCULATED CROSS SECTION FOR NEAR CENTRAL U " U AT ELge " 400 MaV/ NUCLEON o. = iB69 b a = 0 .5

z 0

eL4B

N

I

CALCULATED CROSS SECTION FOR NEAR CENTRAL U " U AT E~AB' i00 MeV/ NUCLEON o: ~ 1.869 b ewB " 30°

= 30°

5 a

w w w w

l00

60°

a W

m 0

10

90° I .,n 0 I I

0

1

I 100

1

I 200

ENERGY OF REACTION PRODUCTS MW /NUCLEON

I 300 0

I 100

1

\I 200

\

ENERGY OF REACTION PRODUCTS MTV / NUCLEON

1 300

Fig. 6. (a) The figure shows the nucleon spectra for various angles for the U+U reaction (calculated again under the assumption of explosion of the total system). (b) A comparison of the B,, b = 30` and 90° directed nucleon spectra for the U+U. reaction, predicted by the present model and those obtained from a moving Boltzmann distribution with no radial flow. The wmmon parameters for the two models were the same : eo = 100 MeV, eß(0) = 100 MeV, oo = 1869 mb. For our model the values a = 0.5, 1 .5 and 3 were chosen . For the other case the relation eß(0) _ (~~cT was used .

Figs. 5 and 6 show the calculated proton spectra of the Fe ~-Ag and U+ U reactions. It should be noticed that since the nucleon cross sections all depend on the parameter b~lto, the dependence of the maximum density vp o cancels. This means that only the shape parameter a remains as a free parameter. It also means that from these cross sections the maximum density is not observable in the model. 6. Conclusioos The hydrodynamic model for an expanding hot spherical gas cloud presented in this per is attractive because of its simplicity . It has been compared with calculations in a MonteCarlo model of head-on heavy-ion collisions. Despite the fact that the two methods approach the same physical problem from very different directions (classical macroscopic versus classical microscopic) and furthermore the fact that the boundary conditions for initiating the hot gas cloud are not spherical in heavy-ion

FIREBALLS

33 1

collisions, there is close similarity between the results of the two methods. The calculated spectra of emitted nucleons show broad peaks which shift to higher energies for more forward angles . Experimental data for energetic large-ion beams are scarce . However, there are indications that they also qualitatively show this feature. This could be regarded as a fingerprint of the presence of a hydrodynamic flow during the expansion of the compressed nuclear matter . Existing experimental data on double differential cross sections of reaction products have so far been integrated over all impact parameters D. In this way, the more central collisions, which are supposed to exhibit the hydrodynamic flow, are mixed with peripheral collisions. We therefore strongly suggest that reactions between big ions are performed in such a way that the near-central collisions can be selected. One expects that such central collisions are accompanied by a large multiplicity of reaction products, and a characteristic symmetry of the distribution of emerging particles in each ion-ion collision event. As a final remark, one can conclude that a case for hydrodynamics, if any, will most likely be found in collisions between big and energetic heavy ions. One of the authors (J.Z.) wishes to express his gratitude to the Niels Bohr Institute for the kind hospitality during part of this work. Appendix BREAK-UP QUANTITIES

The break-up criteria (6b) immediately yields the relations (notations as earlier ; the subscript b stands for break-up)

(A3) ABx(1-x2)~ +~

Tb

Here

_

1-xZ

CAz(1-~)i+~+tô .

(A .4) (AS) (A .6)

33 2

J. P. BONDORF et al.

Here

x = r/R .

(A.9)

The quantities of interest for calculating double differential cross sections are : C(1-xz) mAZBzxz(1-x z) 1 +~ PnRb = a(1-xz~.

(A.12)

Appendix B THE ß-FUNCTION

For the sake of completeness the definition of the ß-function is quoted

this is related to the y-function by

r(x>r(y)

B(x, Y) = r(x+Y) References

G. D. Westfall et al., Phys . Rev. Lett . 37 (1976) 1202 G. F. Chapline et al., Phys . Rev. D~B (1973) 4302 W. Greiner et al., Phys . Rev. Letz. 32 (1974) 741 J. R. Nix et al., Phys . Rev. Lett. 3S (1975) 905 M. I. Sobel et al., Nucl . Phys . A251 (1975) 502 Y. Kitazoe, K. Matsuoka and M. Sann, Prog. Theor. Phya. S6 (1976) 860 K. P.,Stanyukovich, Unsteady motion of continuous media (Pergamon Press, NY, 1960) p. 79 and other 8) J. P. Bôndorf, H. T. Feldmeier, S. Garpmen and E. C. Halberz, Phys. Lett . 66B (1976) 217 9) J. R. Nix et al., Phys . Rev. Lett. 38 (1977) 1055 10) J. D. Walecke, Phys. Lett .'9B (1975) 109 1) 2) 3) 4) 5) 6) 7)