The rarefaction of oblate shaped nuclear matter in heavy-ion collisions

Nuclear Physics A305 (1978) 226-244 ; © North-Holland Publtalriny Co ., Anraterdatn Not to be reproduced by photoprint or microfilm without wrlttea permiuion from the publisher

THE RAREFACTION OF OBLATE SHAPED NUCLEAR MATTER IN HEAVY-ION COLLISIONS J . N . DE', S . I . A . GARPMAN tr and D. SPERBER

f

RPI Troy" , NY 11181, USA J . P . BONDORF The Niels Bohr Institute, University of Copenhagen, DK-1]O(T Copenhagen ~, Denmark and J . ZIMANfI Central Research Institute jor Physics, H-1515 Budapest 114, PO Box 49, Hungary Received 3 January 1978 (Revised 17 April 1978) Abstract : We examine the disintegration stage of a hot zone in a high energy heavy-ion collision using hydrodynamics. We consider the initial non-homogeneously compressed matter to have an oblate shape. An approximate solution to the hydrodynamic equations is obtained by solving two coupled non-linear differential equations for the semiaxes of the boundary . We show that the boundary passes through a spherical shape and finally ends in a prolate configuration as the rarefaction proceeds. We apply the model to heavy-ion reactions to calculate angular distributions. We compare our theoretical results with experiments selected for central collision for the reaction C+Ag and O+Ag at energies about 200 MeV/nucleon . Good agreement between theory and experiment is obtained . Comparison with experiment allows estimating the ratio between the major and minor axis of the original oblate shape . It is shown that both non-homogeneity and shape asymmetry lead to enhanced forward and backward ejection of matter in the c .m . frame.

1. Intradtution Highly excited nuclear matter (with temperatures about 30 MeV) can be formed in heavy-ion experiments with incoming energies greater than 200 MeV/nucleon. Recent experimental data t . z) have been explained with reasonable success using the "hot spot" idea [fireball model, set* ref. 3 )] . In a previous paper a) we studied the hydrodynamic expansion ofa compressed spherical configuration. After compression the zone is supposed to pass through a stage with irrotational flow . The use ofa spherical initial stage for the expansion in ref. 4) was to a certain extent chosen due to the fact that this configuration offers an especially simple solution to the equation of motion . Calculations, both with hydrodynamic models as well as r Supported by United States Department of Energy . tt Supported by the American Scandinavian Foundation and United States Energy Research and Development Administration . 226

RAREFACTION OF NUCLEAR MATTER

227

with statistical microscopic models 9" ts) show that the configuration for central or near central collisions just before expansion should be described by a flat oblate shape (see fig. 1). In the present discussion we are concerned with the expansion of a spheroidally shaped compressed zone. Again after compression there is an initial time t = 0 at which there is no internal flow . The decay of the hot zone is treated using hydro_. .

Fig. 1 . Schematic view of a near central heavy ion collision. A represents the expanding hot zone moving with a velocity V, B representing spectator almost at rest .

dynamics neglecting transport. We now proceed by introducing a new degree of freedom describing quadrupole deformations. We assume the compressed zone to be initially oblate . The initial shape is specified by its volume and the ratio between the major and minor axis. 2. Theory According to the present picture the central or near central reaction is divided into two parts: First a compression stage, for which the momentum flow eventually creates a high density zone tt), which is not considered in the present discussion due to the severe uncertainties of the equation of state for nuclear matter in these early stages of the reactions. Furthermore, quantum effects play a significant role for early times of the compression. Second an expansion stage of hot compressed matter which can and is treated hydrodynamically. In the present paper we focus attention on the second stage of the reaction. The first stage is described in a phenomenological way by two parameters : a form factor and the ratio between the major to the minor axis at t = 0, which is the starting time of the isentropic expansion. During the first stage when the flow of matter is still ingoing the ejection of nuclear matter is limited.

22 8

J . N . DE et al.

Therefore the main contribution to observed angular spectra will be attributed to the later stage. This is in contrast to Mach cone predictions') where shock waves are idealized as propagating in semi-infinite nuclear matter as cones. In that picture the passage of a shock wave through the nuclear surface is assumed to eject particles at definite angles . However, Monte Carlo calculations based on free nucleon-nucleon cross sections 1 Z) show no such sharp shock fronts. The non-existence of sharp shock fronts in heavy ion collision at higher energies was also demonstrated by Smith i 3 ) in Monte Carlo calculations. The reason for this can be attributed to the relatively long mean free paths for the nucleons as compared to the nuclear dimensions. Since the incoming energies which we will consider ( x 200 MeV) correspond to collision speeds which are greater than the speed of sound in nuclear matter "), we will allow the entropy to be a field. Transient shock waves in the first stage are thus included. Recent experimental data for central collisions ia . is) show no sharp peaks in the angular distributions as predicted by strong shock-wave models. We therefore neglect particle emission in the entrance channel. In the present treatment we use in addition to the hydrodynamic equation a a polytropic equation of state. To simplify the treatment further irrotational flow is assumed. At velocities exceeding the speed of sound in nuclear matter the one-body viscosity is not relevant. On the other hand the shearing viscosity term in the NavierStokes equation vanishes identically for irrotational flow . The heat conductivity terms in the energy balance was found to be negligible a). For the hydrodynamic equation a) we use the continuity equation and momentum conservation which are D +p0 ~ V = 0, Dt

(la)

DV mp Dt +VP = 0.

(lb)

Here p, V and P are the lpca! number density, velocity and pressure field respectively. Here the Stokes derivatives D/Dt is defined ô D___ô _ Dt ôt + vs ôx + ar âyy

+ U=

ô_ô _ô V ~ grad. ôt +

ôz

We will assume the local equation of state to be that of an ideal gas such that f =

Pp-Y.

(lc)

Here y = C P/Cv is the ratio of the specific heats. From (lc) we get the additional condition for an isentropic expansion, namely Dl = 0. Dt

(ld)

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229

For an ideal gas the energy balance is

where T is the local temperature. We search for a similarity solution to eqs. (1) using the ansatz P(r, z, t) _

~ ~(r, z, t~, a(t) b(t)

egr I'(r, z, t) = a(t)zrb(t)r ~(r, z, tY` + ar,

(2c)

where a(t)

b(t)

(Cylindrical coordinates are used with the z-direction along the beam.) Here a(t) and 6(t) are the semiaxes of the boundary perpendicular and parallel to the beam respectively, and g, e, a, p and y are constants. It can be seen that a determines the form factor . Combining (2b) with (ld) one obtains u=

râ(t) , a(r)

(~)

zb(t) where u and v are the components of the velocity vector V in cylindrical coordinates Eqs. (2e) and (2f) clearly satisfy (la). The equation of motion (lb) is also satisfied if the following conditions are fulfilled [combining eq. (lb) with egs.(2a), (2c), (2e) and ôu ôz

0'

(3a)

âv 0. ôr

(3b)

[Eqs . (3a) and (3b) and the cylindrical symmetry yields the relation V x V = 0, i .e.,

230

J . N . DE et al.

the flow is irrotational .] We also have ~+ay = a+l,

(3c)

m K a= a z r _ tbr _ t

(3e)

K b - azcrtlbr .

(3f)

The constant K will be referred to as a "shape coupling constant", and its precise physical significance is discussed later. We will only consider the case y = 3 which is the value for an ideal monoatomic gas with translational degrees offreedom only 4). The analytical solutions to (3e) and (3f) (for small times) is derived in appendix A. Formal analytic solutions for all times are derived in appendix D. These solutions

Fig. 2. The time evolution ofa(t) and b(t) which are solutions of eqs . (A.1). Here s u = 3 .86 fm, bo = 1 .29 fm and K = 2 .76 fm~ ~ c~ .

23l

RAREFACTION OF NUCLEAR MATTER a,=3~86 [fm] b,=1 .29 [fm] K "2.76 ~fm2ez~

/

.i

- . ~t=17.97 [fmk]

.~

\

t"7.66 [fm~~.

v

Î

1

. ._~ - ._~~

/.

Fig. 3. The shape of the boundary for time t = 0 (solid line) t = t, = 7.66 fm/c (dashed line) and t = 17.97 fm/c (dot-dash line), for the same ao , b o and K as in fig. 2.

are in the form ofincomplete elliptic integrals. However, for the purpose ofcalculating cross sections it is convenient to use numerical solutions of the differential equations. For all times the equations are solved numerically. Using a one-step finite difference method we integrated eqs. (3e) and (3f). 1'he results ofthe integration for a representative case are shown in figs . 2 and 3. It can be seen that the initial oblate shape of the boundary will pass through a spherical shape and finally end in a prolate shape as the time goes to infinity s). We find numerically that a7/363/3 i s . F(X)

0

K

0

Here t, is the time to reach the spherical shape, ao and bo are the semiaxes at time 0, and K is the shape coupling constant. Here xo is defined by xo = ao/bo.

As can be seen from table 1, F(xo) is a slowly varying function of xo. 3. Application to heavy-ion coWsdons

As the ions approach each other a compressed zone is formed. The evolution of this zone is studied in this paper. The situation in the formation stage resembles the

23 2

J. N . DE et al. T~st .~ 1

The function F(xo) xo

F(xo)

0.1 0.5 0.9 1 1 .0 2 10

3.2 2.1 2.0 [0] 2.0 2.0 2.7

collision of two supersonic jets te) . If target and projectile are of equal or almost equal size this compressed zone has an oblate shape t t " t z. t e). For cases in which the target and projectile masses differ considerably the shape of the compressed zone is uncertain. For intermediate difference in size the assumption for a oblate compressed shape is reasonable . Since, in the cases considered in this paper, the projectile is smaller than the target it will pass in a supersonic way through the target and will interact only with the region of mutual overlap 6). In this case the nucleons in the overlap region come to rest producing a compressed hot zone t 1). The constants e and g in eqs. (2a-c) can be related to the initial heat energy Eh (0) and number of particles N in the hot zone by 2a

J

CP(r, z, t = 0)rdrdz = Eh(t = 0),

(6a)

2n p(r, z, t = 0)rdrdz = N.

(6b)

J

(For a more detailed discussion of energy conservation see appendix B.) We get _ Eh(O~â/ 3 b/3(2+a)2 ~ s/s 1 3+a)3 ' 9 ~ 9=

N(a+1)

a

1

B(Z, 2 + a)'

where B(x, y) is the normal beta function. Combining eqs . (6cß (6d) and (3d) we get _ 4/3 2/3 Eh(t - ~)

Here m is the nucleon mass inside nuclear matter (931 MeV). Rewriting in terms of the initial volume : V° _ ~b°aô and the total mass MT = mN: K=

V Eh (t °

MT

0) a + ~3~~. 2) ~

4n

(6f)

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233

To perform a specific calculation of cross sections (for details see appendix C) we use the following assumptions : (i) The number of participants e) in the target is determined from the volume swept in the target by the projectile . (ü) All the kinetic energy available to the participants in their c.m . frame is converted to heat . From these assumptions we get N1 2e~,b ~ (units of c), V`.~. - N I +N Z (mc 2 ) N 1Nz

(7a) (7b)

Here V~.m. is the c.m. velocity of the hot zone : e~ ¢b is the incoming kinetic energy per nucleon and e b(t = 0) is the heat energy per nucleon at t = 0. Here N 1 and N Z are the number ofnucleons in the projectile and the number of participating nucleons in the target, respectively. The geometrical and dynamical assumptions allow us to determine V~.~. and eb. By keeping these parameters fixed the angular distribution depends only on two free parameters : xo and a. A comparison between calculated and measured distributions for central or near central collision allows us to estimate these parameters . The angular distributions are found to be independent of the initial volume Vo (the initial volume is not an observable in the model). For simplicity we choose Va to be the volume of the smaller nucleus. The observed angular distribution is due to an inclusive spectrum rather than individual nucleons l '). To simulate the clustering into d, 'H, 3 He and ' 1 Be and heavier nuclei we chose a finite detector distance R. (See appendix C.) This distance was chosen so that the main contribution to the cross sections is due to events before the total heat energy has reached the deuteron binding energy. We calculated the angular distributions for the inclusive reactions O+Ag and C+Ag for a number ofdifferent xo and a. The results for O+Ag for this calculation are shown in figs . 4--6 . Experimental data for these reactions at energies about 200 MeV/nucleon are displayed in fig. 7. 4. Results and diacossion 4.1 . EXPERIMENTAL DATA ON CENTRAL COLLISIONS

The experimental data for selective central collisions are relatively scant. Photographic emulsion data of Otterlund et aI. s) for the inclusive reaction O + Agar have recently been reported. Also track detector experiments for the inclusive reaction C+AgCI have been performed earlier by the Frankfurt group'). The data are in

234

J . N. DE et al.

Fig. 4a . The angular distribution da/dA (in the lab frame of reference) for Yo = 107.3 fm', v~ .m. = 0.18c, eti =40MeV,a= 1,R=6fmandzo = 1 .

80 A~J Fig. 4b . The angular distribution (1/2n~o/dB for the same set of parameters . qualitative agreement though they differ in finer details. In both cases t~lte central collisions were selected out according to the multiprong hypothesis. The spectrum from the Frankfurt group as shown in fig. 7a shows a narrow peak at a forward angle around 40° (for incoming energies 250 MeV/n) and a rather broad "platesu~ at 80°

RAREFACTION OF NUCLEAR MATTER

235

dc

âa

(aul 0.24 0.20

alb alt Q08

01

1

o

zo

1

0o

I

1

eo

1

eo loo e Ub ~'~

1

eo

1

wo

1

I

Iso

180

Fig. Sa . The.same as fig. 4a but a = 10 - ' and xo = 3 (dashed curve) and a = 4 and xo curve) . s~a9

=

3 (dot-dash

cb)

(a .ul 0.1

/ / /

aos

/

\

/

.\

\ ~\ \\ \

I/

.\

I/

\\ \ w '\ \ ' \ .~\

'J

~l/ /

s\

'~/ °o

\~~ .

zo

ao

so

eo

loo

¢o

wo

Iso

leo

Fig. Sb . The angular distribution (ll2rz~aldB for the same set of parameters . to 130°. In contrast the Otterlund results (fig . 7b) only show a broad peak at about 60° (incoming energy 200 MeV/nucleon) and without any plateau. The source of the experimental inconsistency might lay in the fact that in ref.') nearly pure C+Ag collisions were encountered, due to the following reasons :

236

J. N. DE et al .

80

100 6~,~ C"1

Fig. 6a. The same as fig. Sa but a = 1, xo = 3 (dashed curve) and a = 1, xo = 5 (dot-dash curve).

BO I00 A B C]

Fig. 6b. The angular distribution (1l2n~a/dB for the same set of parameters. (i) The multiprong hypothesis favorably selects C+Ag collisions as compared to C+CI collisions . (ü) The ratio of the reaction cross sections :

RAREFACTION OF NUCLEAR MATTER

237

9 ueC"] Fig. 7a . Experimental inclusive angular distribution dNldB [ref.')] solid curve) as compared to theoretical angular distributions (dashed lines) : V = 80 .5 fm', V~ .m _ = 0.19c, eh = 48 MeV, a = l, xo = 3 (dashed curve) and xo = 5 (dot-dash curve).

Fig. 7b . The same as fig. 7a except that experimental results are taken from ref. s), and V = 107 fm' V~ .m . = O.18c, eb = 40 MeV, a = l, xe = 1 (dashed curve) and xo = 3 (dot-dash curve) .

In contrast in the work in ref. e) with an Agar target the reaction O + Br will be nearly as likely as the reaction O + Ag since ~s~

_ (~)~ = 1 .23.

23 8

J . N . DE et al.

Due to the above considerations it is clear that the mass asymmetry is different in the two experiments (defined as the ratio between projectile and target mass): 0.11 0.15 0.20

for C+Ag ref.') for O +Ag( x 60 ~ of the cases) ref. e) for O + Br( ~ 40 ~ of the cases) }

One can speculate that a higher mass asymmetry will also lead to a high shape asymmetry. However, to confirm such a speculation more experimental results are required . 4.2. THEORETICAL PREDICTIONS AND COMPARISON WITH EXPERIMENTAL DATA

We calculated the angular distribution for the inclusive reaction O+Ag for a number of different x° and a (for completeness we include the spherical case or x° = 1). The results of calculations are shown in figs. 4-6. In fig. 4a we display the angular distribution dtr/dlil for a spherical shape x° = 1. It can be seen that in this case the cross section decreases monotonically with increasing angle. For x° = 1, dQ/dB has almost a sinusoidal shape. (This statement is exact for V~.m. = 0.) These statements are independent of the details of the form factor . On the other hand a very high value.of x° predicts (for dQ/dB) a wide plateau at back angles . This is due to the enhanced ejection of matter in the forward and backward directions (in the c.m .). Our result is consistent with the results of Kitazoe et al . ' °) at lower energies and densities. However, the discrepancy for higher densities and energies can be attributed to the constant density and linearization of the equations in ref. 1°). The work of Nix 9) indicates that for collisions with intermediate impact parameters (for which x° is close to 3) one obtains an increase in the cross section at backward angles similar to the increase obtained in the present model. Comparison with the experimentally determined width and position of the plateau in the angular distribution du/d8 allows us to estimate the parameter x°. As can be seen from 6gs. 5 and 6 an increase in either. of theIparameters x° or a will lead to enhanced forward and backward ejection of nuclear matter . Inspecting the representative cases we can conclude that the angular distribution is more sensitive to the quadrupole deformation x° than to the form factor a. A Monte Carlo calculation assuming that the nucleon-nucleon cross section equals to the free nucleon-nucleon cross section suggests .a value of the form factor around unity 4). We calculated the angular distribution for a = 1 and x° = 1, 3 and 5 as shown in the figures. For a mass asymmetry of0.11 we extract x° = 5, whereas for an effective mass asymmetry of 0.17 we extract 1 5 x° 5 3. Even in the event of clustering the present model for angular distribution is applicable since the cluster moves along the macroscopic flow direction. The present status of the experimental data does not allow an accurate determina-

RAREFACTION OF NUCLEAR MATTER

23 9

tion of the parameters involved. However, the determination of the position of the plateau and the ratio of the cross section at backward angles to the cross section at the plateau will make the present analysis more accurate . It is possible to generalize the present theory to non-central collisions. However, in this case there will be a change in (a) the number of participants, (b) the c.m. V~.~. and (c) in heat energy per nucleon eb [ref. s )] . It is difficult to apply the theory, in its present form, to reactions of high mass asymmetry since in that case the projectile might be stopped in the target and the underlying assumption of the present model may not be valid . The model can be generalized so as to be applicable in the relativistic region. The dynamics in the c.m. will be unchanged. However, the relativistic transformation between the laboratory and c.m. systems of coordinates must be used. Appendix A ANALYTICAL SOLUTIONS OF THE DIFFERENTIAL EQUATIONS (SMALL TIMES)

We want to solve the coupled non-linear equations : K Q7/3bz/3'

(A.la)

K ~ - a4/3bs/3 ~

(A.lb)

a

We use a Laplace transform for the equations remembering â(0) and li(0) = 0 (due to time reversal symmetry) â(s) -

6(s)

K~

0

= K~

0

e-~a(t)-,/sb(t)-2/3dt+

s° ,

e-~a(t)-a/3b(t)-s/3dt+ b° .

(A.2a) (A.2b)

Here the â(s) and b(s) are the Laplace transform of a(t) and b(t). We expand a(t) and b(t) in even powers of t : ~ aZ"t2"~ t) c "=o and define the functions :

a-7/3(t,~-Z/3(t~ F(t) c G(t) = a-ala(t~-sl3(t) .

(A.3a)

(A.4a) (A.4b)

240

J . N . DE et al.

The solution is then given by K a° K K .. .. . . â(s) _ + -s F(0)+ s F(0)+ ~ F (0) . . ., s s s s

K

K G(0)+ K $(s) = b° + G(0)+ ~G~(0) . . .. (A s s s s

(A.Sa) .Sb)

By truncating the expansion (A.3a) and (A.3b) to lowest order (n = 0 only) eqs. (A.4}-(A.5) give the next order expansion ccetïicients : azbz. A recursive usage of eqs. (A.3}-(A.5) allows determining the expansion coefficients to any order. The lowest order expansion cce~cients are a0 -

az

~~)'

K a71362ß2' 0

0

z

aa

72aö sbôß {QÔ + bô}' _

ae

K3 259 106 40 405 x 16aôbô { aô + aôbô + bô}' K bz - a4ß65ß2'

bo = ~0)'

I

0

L

2

72a~ b /a

ba _

be

0

405

{â

+ bô}'

K3 112 148 145 16a 463 { a4 + azbz + b40 } ~ 0 0 0 0 0

X

(A.6)

The expansion is only valid for small times (the exact convergence radius for time being a function ofa°, b° and K). This is useful for initiating the numerical integration. Appendix B THE ENERGY BALANCE

The thermal energy of the gas cloud at any time is given by E~(t) = 2n

J

Cvp(r, z, t)T(r, z, th^drdz.

(B.1)

Here T(r, z, t) (MeV) is the temperature field and Cv is the specific heat at constant volume. In our case Cv

=r

RAREFACTION OF NUCLEAR MATTER

241

For an ideal gas (B.1) can be rewritten as Eh (t) = 2rz

C~P(r, z, t)rdrdz.

(B.2)

p(r, z, t){uZ(r, z, t)+v2(r, z, t)}rdrdz.

(B.3)

The flow energy Ef is given by Ef(t) = mn

J

J

The energy balance requires [Ef(0) _- 0] Eb(t) +Ef(t) = E h(0).

(B.4)

Let eb by the heat energy per nucleon, then eqs. (2a), (2c) and (B.2}{B.4) yield

but according to eq. (6e) this can be written 2d2 +üZ = 3K{

1

4/3b2/3 l((a0 0

-

1

a 413b2/3

'

(B .6)

This is a first integral of motion . It is equivalent to eq. (4.7) in the work of Nemchinov s) who used a Lagrangian formulation. As can be seen when t -~ o0 the system cools down to zero temperature and the energy can be attributed to flow energy only, 2â2 (oo)+1i 2(oo) = 3K/(aô/sbô/ 3) .

(B.7)

Appendix C ANGULAR DIFFERENTIAL CROSS SECTIONS

We define the local particle current as Here E, and E= are .unit vectors in cylindrical coordinates . A unit vector in the radial direction is rP,+ z2Z 8x (C .2) r +z The vector area element in polar coordinates is

dS = RzPRdIl .

The instantaneous rate is given by

(C .3)

242

J . N. DE et al.

By integrating over time we get the number ofparticles per unit solid angle element as dN

d~

°° r +z ~ (az+ur)pdt.

=

(C.5)

0

The angular distribution P(R, B, ~) is given by

where N is the number of particles in the hot zone . Obviously the angular distribution is normalized to unity so that

Finally the angular differential cross sections are given by dtr dSè

where

Q~

lim Q P(r, B, ~), x~~

(C.8a)

is the total reaction cross section for formation of the hot gas cloud. Appendix D

THE GENERAL SOLUTION OF THE DIFFERENTIAL EQUATION FOR A SPHEROID

Eqs . (3e) and (3f) for y = 3 are K aa = aa/3bz/s'

(D.la)

K bb = aa/3bz/a

(D.lb)

And from eqs. (B.5) and (B.6) for fixed a we obtain {2 aa/ bz/s +zbz+az}

bz/3' 2 aa/ 0 0

which can be written as [using (D .la) and (D.lb)],

(D.2)

RAREFACTION OF NUCLEAR MATTER

243

Integrating over time we get ad+Zbb = 2 a4/bz/a t . 0

(D.4)

0

where we also used that â(0) = 6(0) = 0. Integrating over time again we get the phase trajectory : z

az+Zbz = ~oKb ô/a +aô+ibô.

(D .5)

Here both a and b go to infinity with time and the motion is unstable with a saddle point at (ao, bo). We introduce constants C and D such that C= D

3K

(D.6)

2a 4/sbz/s' 0

0

= aâ +Zbô.

(D.7)

We now use eqs. (D.4) and (D.2) : (b~)z -- 4(ct-aâ)z = bz {2C - 2âz-

(D .8a)

a436z / 3} ,

(D.8b) Using eqs. (D.5) and (D .8b) we get ~

â = Cta~~~ Ct +D-a _ /CDb = Ct6~~~ Ct +D-Zb

3K(C~+D)

(Ctz+D),

(D.9a)

3K(Ct +D) }'(Ctz+D). {b(Ctz+D-Zbz) }

(D .9b)

{16a4(Ctz+D-az)}~'

2CD-

Here the minus sign corresponds to xo Z 0 oblate initial shape. We will change both the dependent and independent variables so as to be able to separate eqs. (D .9a) and (D.9b) :

(D.IOd) dq _ s dS z JJD/6oq (q - )2CD-3Kq {4 - z} - J~n CS S -D~ 4

(D.lOe)

244

J. N. DE er a! .

As can be seen the left-hand sides of eqs. (D.lOd) and (D.lOe) are incomplete elliptic integrals whereas the right-hand side can be expressed in elementary functions. References

1) J. C. Papp, Thesis, Berkeley (1974) 2) H. H. Gutbrod, A. Sandoval, P. J. Johanson, A. M. Poskanzer, J. Grosset, W. G. Meyer, G. D. Westfall and R. Stock, Phys. Rev. Lett . 37 (1976) 667 3) G. D. Westfall,J . Gosset, P. J. Johanson, A. M. Poszkanser, W. G. Myer, H. H. Gutbrod, A. Sandoval and R. Stock, Phys . Rev. Lett . 37 (1976) 1202 4) J. P. Bondorf, S. I. A. Garpman and J. Zimànyi, Nucl . Phys. A296 (1978) 320 5) I. V. Nemchinov, J. Appl . Math . and Mech . 29 (1965) 143 6) J. D. Bowman, W. J. Swiatecki andC. F. Tsang, Lawrence Berkeley Laboratory Report No . LBL-2908 (TID-4500-R61) (1973), unpublished 7) H. G. Baumgardt, J. V. Schott, Y. Sakamoto, E. Schopper, H. Stocker, J. Hofmann, W. Scheid and W. Greiner, Z. Phys. A273 (1975) 359 8) B. Jakobsson, R. Kuhberg and I. Otterlund, Cosmic Ray Physics Report LUIP-CR-76-04 (1976) Lund, Sweden 9) A. A. Amsden, F. H. Harlow and J. R. Nix, Phys . Rev., to be published 10) Y. Kitace, K. Matsuoka and M. Sano, Prog. Theor. Phys . 56 (1976) 860 11) M. I. Sobel, P. J. Siemens, J. P. .Bondorf and H. A. Bethe, Nucl . Phys . A2Sl (1975) 502 12) J. P. Bondorf, H. T. Feldmeier, S. Garpman and E. C. Halben, Phys. Lett . 65B 13) R. K. Smith and M. Danow, Proc . Meeting on heavy ion collisions, Fall Creek Falls, Tennessee(1977) 14) B. K. Jakobsson, R. Kuhberg and I. Otterlund, Nuel . Phys . A276 (1977) 522 15) H. Jakobssen, Symp . on high energy reactions in Nuclei, Spatind, Norway, January 1977 16) J. P. Bondorf, J. P. Siemens, S. Garpman and E. C. Halbem, Z. Phys. A279 (1976) 385 17) R. Bond, S. Garpman, J. Johansen and S. Koonin, Phys . Lett . 71B (1977) 43