Ground state correlations in hard heavy ion reactions

2.N

I

Nuclear Physics A312 (1978) 149-159; ( ~ North-HollandPublishino Co., Amsterdam

t

N o t to be reproduced by photoprint or microfilm without written permission from the publisher

GROUND STATE CORRELATIONS IN HARD HEAVY ION REACTIONS J. P. BONDORF, G. FAI * and O. B. NIELSEN

The Niels Bohr Institute, University o]' Copenhagen , DK 2100 Copenhagen O, Denmark Received 19 June 1978 Abstract: The possibility of observing neutron-rich isotopes in hard heavy ion reactions is studied. The

influence of target isospin correlations on participant and spectator mass and charge distributions is calculated within the liquid drop model. Calculated spectator distributions for a Pb target are compared to experiment.

1. Introduction

Recently there has been growing interest in what purposes hard heavy ion reactions can be used for l - 5). Most of the experimental investigations of nuclei far from the beta stability line have up to now been done on the proton-rich side with fast protoninduced reactions. For the time being the main sources of heavier neutron-rich nuclei are products of fission and deep inelastic heavy ion reactions at low energy. The division of the reaction products of fast heavy ion reactions into hot "fire ball" material and much colder spectator fragments is a new feature of these reactions which makes it worthwhile to study mass and charge distributions from such reactions. The very fast removal (abrasion z)) of a piece of the target in hard heavy ion reactions offers a new possibility of investigating certain features of nuciear ground state correlations. By very fast abrasion we mean that the reaction time is considerably shorter than the time necessary for considerable amount of disturbance to penetrate deeply into the two nuclei. Taking the speed of disturbance to be at least of the order of the Fermi speed this leads to bombarding energies above of the order of 50 MeV. The fast abrasion results in a relatively clean cut between different subvolumes of the nuclei in the reaction. This is the picture underlying the participant spectator division. In the present paper we examine the effects of isospin correlations in the nuclear ground state on the observed mass and charge distributions of fire ball and spectator products. The observed products are not the primary fragments left directly after the collision, but rather the secondary products emerging from the decay of the primary ones, as discussed in ref. l). Another purpose of the present investigation is to see to what extent characteristic ground state features of the primary spectator distribution survive the decay. On leave of absence from the Roland E6tv6s University, Budapest. 149

150

J.P. BONDORF

et al.

The general geometry providing the framework of our calculations is described in the fire ball 3), rows on rows 4) and fire streak 2) models. We will utilize the latter picture throughout this paper. We have in mind collisions in which there is a partial overlap of the target and projectile with an impact parameter b [ R x - R p I -< b -< R x + R p ,

(1)

where R-r and Rp refer to target and projectile half-density radii, respectively. The volumes vp and v-r which are outside the overlap region contain the two spectators. 2. Ground state correlations

We are interested in the distribution of neutrons and protons in the fire ball and and the primary spectators. In an independent-particle model such distributions have been given in ref. 2). Each of the two nuclei is divided into a spectator and a participant part. The two participant parts mix into a common fire-ball-like structure. In order to find the final distributions of mass and charge in the spectator and participant parts, we first consider a spectator volume v with a sharp boundary in one of the nuclei. The precise identity of the spectator is given by the dynamics and will be specified later. The isospin potential, however, modifies the charge and mass distribution considerably, since it puts a constraint on the system, which prevents very large local differences in the densities of neutrons and protons, respectively. In the following we discuss an approximate method by which it is possible to estimate the modification of the distributions caused by the isospin potential. We want the probability of finding exactly n neutrons and z protons in the nuclear spectator subvolume v at the time of impact. We assume a sudden approximation in which the nucleon distribution in the spectator remains unchanged during the supposedly very short collision time. In this picture the density fluctuations in the ground state may give characteristic nucleon distributions for the spectator which we now estimate. We use a fluid-dynamic picture of the nucleus to calculate the probability distribution starting from the isovector fluctuation density pl(r, t) = (~(/On(r, t)--pp(r, t)),

(2)

where Pn and pp stand for neutron and proton densities, respectively. Thus P i describes the fluctuations in the difference of neutron and proton densities, or, in other words, the vibration of the neutron fluid with respect to the proton fluid (polarization modes). These vibrations can be ,treated in several ways 6, 7), but in order to get a rough estimate we chose the simple hydrodynamic model of chapter 6 in ref. 7). In this case the general solution and the boundary conditions have the form

HARD HEAVY ION REACTIONS

151

pa(r, t) = Po ~ (-- )~x/22 + lj~(k.~r)[ Y~(O, ~o)ct.~(t)]o, .zu __~r ~ (Ja(k"xr)) r=R

=

O.

(3)

Here Po is the total particle density, 2 is the multipole order, k,x the wave number, ja and Ya denote spherical Bessel functions and spherical harmonics, respectively, while [ ]o stands for angular momentum coupling. The operator ~,a(t) can be expressed in terms of the boson creation and annihilation operators c,x and c+~ as a,;~(t) = (0~o),z(cL + c,x),

(4)

with (~0),z = X/2C,a the zero-point amplitude for the mode given in terms of the energy ho~,a and the restoring force parameter C.x. We define the non-local spatial fluctuation density #l(v, r') as

#2(r, r') = (Olpl(r, t)pl(r', t)[0),

(6)

where [i)> stands for the correlated ground state 7). The average of (6) is calculated for an arbitrary sub-volume of the nucleus to obtain the width of the distribution inp f l t" l½ 00, = ~.~-/#Z(r, r')d3rd3r'~ . ~,v j~ )

(7)

In (7) we take the subvolume v bounded by a sharp surface in the spirit of(3). Exactly the same procedure can be followed to calculate the fluctuations in the total density

p(r, t) = p,(r, t)+ pp(r, t).

(8)

The only difference is that the compression modes obey a different boundary condition. ja(k'.zr)lr=R = 0.

(9)

Starting from (8) and (9) we calculate 0o, the width of the distribution in the total density. We define width constants k o and kp, by

t~° = kpp°A-~'

(10)

ap, = kp,po A - ~ Results for k o and kp, are shown in table 1 for some geometrically simple cases. We get converging series in the multipole order 2 and the radial quantum number n. The convergence has been tested by going up to 2 = 9, n = 13. It should be noted that, as long as the subvolume considered is sufficiently different from 0 and the total volume, the ratio ap/ap, is approximately constant, with a value of 2-2.5. This can

152

J . P . B O N D O R F et al. TABL~i l

Width constants kp and k v , (10) and the ratio a~/'a,, = 2 k v / k p , for different subvolumes v as given by the integration lir~its in the nuclear center polar coordinates r. 0 and q~ Limits of integration

r~ r2 01 02 4h 4~2

0

t!,' V

//max

k~,

k~,,

20-O/'O-pl or

Ga,"cYl

Rv 0

n

0

n

½RT Rv 0 0 Rs 0

n n

0 0

n ~n

~

~R~ R1 0

n

0

~n

72

0

~n 0

~n

Rv 0

"~max

3 5 9 9 9 9 9 9 9

5 5 9 13 13 13 13 13 13

1.192 1.227 1. 116 1.194 1. t 39 1.033 0.987 0.747 0.697

0.475 0.491 0.534 0.536 0.520 0.493 0.468 0.372 0.347

5.02 4.92 4.18 4.45 4.38 4.20 4.22 4.02 4.02

The numbers "~max and gtmax refer to the maximum values of the multipolarity and number of nodes in the expression (3).

be understood remembering that the relevant physical quantities detei mining a , and a~,~ are the compressibility coefficient beomp and the symmetry coefficient h~ .... respectively. The ratio cr,/a,~ can be cast into a form c;;,//¢71~ ~ (b~ym/b .... p)~G, where o f the poor, MeV,

(l l)

G is a geometrical factor, which is different for the two modes only because different b o u n d a r y conditions. (Our knowledge o f especially bcompiS rather but for the present estimate we use the values bsym ~ 50 MeV, bcomp ~ 15 ref. v).) W e get from (11) o-~,/o-f,~ ~ 2.5G.

F r o m table 1 we see that for those subvolumes which have been investigated, G is not very different from 1. In the following the variable t 3 = ½ ( n - z ) and a = n + z are used instead o f the densities such that

(~"t, = l~'~pl"

(12)

It is rather complicated to perform the integral in (7) for the actual spectator for any impact parameter. We therefore approximate the spread (~a) in the total number a of particles in the subvolume v by the independent particle model estimate [2] as o-. =

A.

(13)

We then use the results for the smallest subvolume in table 1 to obtain for the spread

HARD HEAVY ION REACTIONS

153

in the third component of the isospin O't3 ~

0.24a a.

(14)

The actually used subvolume v in eq. (13) is defined dynamically as described below. It should be noted that the dependence of the fluctuations on A is different in the hydrodynamic and the independent particle descriptions (A ~3and A ~, respectively). The true dependence on A lies probably between the two powers since the two models represent two extreme pictures of nuclear structure. More sophisticated models, including both particle and collective degrees of freedom, may be used to improve the above results. 3. Reaction dynamics The dynamics of the reaction considered is described within the firestreak model 5). For a fixed impact parameter b each reaction row 4) or tube 1) is characterized by a number of participants and the inelastic excitation energy per particle e in the tube (measured in the frame of the spectator). The Coulomb repulsion between the two heavy ions is neglected for the high incident energies we deal with (from 50-100 MeV to several GeV per nucleon). To determine which nucleons (n + z = a) belong to the spectator, we use the prescription of eascaping tubes 1): the target spectator consists of all those tubes which have an energy per particle less than a critical value ec. It is thus assumed that all those particles which have an average energy larger than ec escape immediately at impact. These particles are now included in the fireball:like assembly of particles. In our model ec is specified to be the binding energy per particle (of the order of 8 MeV). This determines the spectator volume v and the average excitation energy of the spectator E 0. Of course, there is a whole distribution of excitation energies in the spectator around Eo, ranging from Emin, determined by the bulk binding energy of the spectator in the initial shape of v to high energies far beyond E0. The distribution has been approximated by a Gaussian, with a maximum at E0 above Emin and a width fie as given by E o ~ ~ nigi,

i~R+

(15)

ieRT

where ei is the excitation energy per particle and ai is the spread in the energy of a single particle in tube i with number of particles n~, while RT refers to the set of retained tubes (spectator tubes) 1). In the model we have used the estimate a~ = Cei,

(16)

where C -- 1 is our guess for a statistical energy distribution. A few typical examples of E*m~n, E and an as functions of impact parameter b are given in table 2. It is seen

154

J . P . B O N D O R F et al. TABLE 2

Typical examples of Em*i., E and trE of the target spectator in the reaction ~60 + 2°8pb at beam energy E~..... = 400 MeV/N as functions of impact parameter b Excitation energy . . . . minimal average E*~,(MeV) E(MeV)

Impact parameter b(fm)

5.0 6.0 7.0 8.0 9.0

39.5 23.6 12.4 4.5 0.3

Energy dispersion oE(MeV )

147.9 131.6 111.8 91.7 70.6

24.4 22.8 20.7 18.4 15.7

that for lower impact parameters both the E*mi n and E" are rather big, which means many evaporated particles. We describe the statistically generated primary mass and isospin distributions by Gaussians with the special widths given by (13) and (14): Fpri m

.tb, E . . . .

[~ t3) __ [~'

1

exp f

21Z~TarY13 -

(a-ao) 2 2%2

(t 3 ~ f30)2"~

2Cr2~ j ,

(1 7)

where a0 and t3o stand for the mean values of a and t3. The total primary distribution including the energy distribution which we take as independent of (17) is now " .... (a, t 3,E*) = (prim gpy]m ,]b, E b e a m

[a ~ '

t3)N exp {

( E *2a -- 2E)2];" j

'

E * > Emin,

(18)

where N normalises the energy distribution. Because of baryon number and charge conservation the distribution of participants is intimately related to the distributions of the two spectators. The fireball-like participant assembly has the following number of nucleons p = Al--al+Az--a2+sl+s2,

(19)

where the numbers s~ and s 2 refer to the nucleons which escape immediately after impact from the spectators as discussed earlier. For the isospin t3p a definition analogous to (19) holds. The fluctuations of p and t3p each contain contributions from four sources. Although the fluctuation in the fast escape numbers cannot be neglected in a more precise calculation, the main contributions to 6p and 6t3v come from the ( A - a ) ' s and the (T3 - t3)'s. Therefore the ratio at~p/ap for the participant assembly is probably somewhat below the value given in (14) for the spectators. But we expect that it still shows considerable isospin correlation. So by measuring multiplicities of fast neutrons and protons event by event, and

HARD HEAVY ION REACTIONS

155

in coincidence, one expects to observe a smaller %p/% than that of the independent particle model. By fast particles we mean here particles with a velocity larger than, say, the Fermi velocity. Also coincidence measurements of fast protons and neutron carriers such as d, t, 3He, 4He, etc. are expected to show a similar behaviour. Some further loss of correlation is, however, expected for these channels because of the dispersion in the formation process for composite particles. The excitation energy E* is removed from the primary spectator products by particle evaporation or fission. In general the decay is very complicated to calculate~ since there are many primary isotopes, energies and many decay steps with a multitude of branching possibilities. With respect to decay there is a large difference between light and heavy spectators. Light nuclei decay by both n, p and ~'s with often similar rates which means that the mass and charge distribution is to a large extent displaced along the line of fl-stability. There is a little contraction in the 13 direction which can, for example, be seen in ref. 8). On the other hand heavier nuclei decay mostly by neutrons. As a consequence the distribution is displaced in the N direction and rather strongly contracted in t 3. Since data on heavy spectator distributions are readily available, we have restricted ourselves to the decay of heavier nuclei. Decay by fission leads to mass spectra which are to a great extent separated from those resulting from particle decay. So fission will be taken into account only by lowering the primary cross-section distribution. We now turn to the decay model with all its simplifications. Only neutron or proton decay with no branching will be allowed. That means that any excited product in the decay chain will decay according to the conditions: neutron decay if

Sn < Sp + Vp,

proton decay if

Sn > Sp + Vp,

(20) where Sn and Sp mean neutron and proton separation energies, respectively, (calculated from a mass formula 9)) and Vp denotes the.proton barrier. We assume that the decay leads only to the most probable energy in the daughter nucleus, from where the subsequent decay starts.When the excitation energy in a decay product is less than both S, and Sp + Vp, the decay chain terminates. The angular momentum of the decaying nucleus is not taken into account. With the above simplifications it is possible to treat the large numbers of decaying systems required for these reactions within a reasonable computer time. The cross section, differentiated with respect to the impact parameter, is now equal to da

d b (b, nf, zf) =

2nbj~,s .E. ... . .d. (nf, zf)

(21)

with . . . . . . d" Zf)" : J/~,Ebeam[nf,

~-" ~prim ~ | ~dE *',~b, Ebeam[ta i' ÷ L3i' ait3iqd

E*)D(ai,

t3i ,

E*

~

nf,

zf),

(22)

60

70

90

....

/"

I

I

I

90

J

~

,

i

]

100

,

I

,

,

I

110

I

i

60

70

-

, I00

90

/ /

l lO

/

~o~.,~,~

NEUTRONS

100

110

SECONDARY

//~~-6-8

90

""-T--7-~'-~ /

,,oo~,~,,,

-6-~zzzz

110

, ~ . , , , .

I00

~~SECONDARY

/

<.,

70

90

90

,,,

100

I00

....

110

110

/

~o~-,.

CORRELATED, EIL~T'/C/~~_~~~ JJ~R-~ )/ I 70~ F-I~NPART, I5\:>' DEPEND' L\'# ~.qx~ I UNREALIST' 1~7~ /~

Fig. 1. Contour lines for primary and secondary population distributions for the reaction t ° O + 2°spb at beam energy Eb~am = 400 M e V / N and impact parameter b = 7 fm. The contour lines are labelled with the logarithm of the relative probability (arbitrary units).

¢I

o~0

I--0¢Y

70

MASS DISTRIBUTIONS FOR FIXED b

©

0 Z

:

~h

HARD HEAVY ION REACTIONS

157

where D(ai, t3i, E* ~ nf, zf) is the probability that the p r i m a r y nucleus with an excitation energy E* decays to nf, ze. T h e further decay to the g r o u n d states goes via 7-rays without altering c ..... d ~n z ft"~

(22)

.]b, E b e a m ~ f~

This m e a n s that our secondary distributions are terminated before fl-decay, which has not been included in the decay calculation.

4. Results T h e calculated p r i m a r y and secondary distributions for the reaction 1 6 0 - ] - 2 ° 8 p b at b e a m energy E b = 400 M e V / N and impact p a r a m e t e r b = 7 fm 'are shown in fig. l a. The corresponding distributions for the independent particle model are presented in fig. 1b and for c o m p a r i s o n we also include a calculation with an unrealistic p r i m a r y distribution in fig. lc. T h e g r o u n d state correlation gives an elongated p r i m a r y distribution along the N = Z line. This elongation partly survives the decay as seen in fig. l a by c o m p a r i n g it to l b and lc. |n order to d e m o n s t r a t e the effects o f correlation in the secondary spectator distribution we have shown some cuts in the distributions at constant secondary mass n u m b e r (fig. 2). It is seen, that the width ~ c is smaller for the correlated g r o u n d state case than for the independent I

I

1

t,O Z O

*~ 0_ 5

""

Z

o, < Z

o_ I--

31 O

20

30

40

N-Z

Fig. 2. Cuts along a = 162 in the secondary distributions shown in figs. la, lb and lc. (arbitrary units). Full curve with dots: correlated (fig. la); full curve with crosses: independent particle (fig. lb); dotted curve: unrealistic (fig. 1c).

158

J . P . B O N D O R F et al.

particle one. At the point where the probability is ~0 of the maximum for a = 162 we find that for b = 7 fm the calculated full width in t 3 is 3.1 for the correlated and 4.0 for the uncorrelated case. The rather small difference between the widths shows that one loses information by the decay process and therefore it will be difficult although not impossible to distinguish between the models by looking at different secondary spectator distributions. An improvement would be to design experiments in which the impact parameter could be selected. Also light spectators should be considered. On the other hand the participant distribution does not suffer strongly from the problem of decay and we recommend participant correlation experiments.

i

80

i

i

i

C R O S S SECTION C O N T O U R S FOR Pb - TARGET

T1 0 3

,

i

~ " ~ 101 "1 ~

,

'~

Z S ~
SPECTATOR

70

/////

i 70

i

i 80

;

i

z~"

i

90

I 100

i

I 110

I

i 120

NEUTRONS

Fig. 3. Cross-sectioncontour lines for secondary products of the target spectator. It is indicated how the centroids of the secondary masses correspond to the impact parameter. The experimentalcurve is from ref. 10). In fig. 3 the total cross section integrated over impact parameter is given and compared to the results of experiments lo). The facts that in the experiment 12C is used as projectile and the beam energy is 2.1 GeV/N, are not expected to influence strongly the mass and charge distribution of the spectator. In fig. 3 we also indicated how the impact parameter b is correlated with the secondary mass. Since RT + Rp < 9 fm in the calculated example, the theoretical distribution in fig. 3 has not been extended beyond this impact parameter. Thus the target-like secondary products are outside the reach of the model. Throughout this paper we have discussed the cross sections integrated over angles. We have not calculated angular distributions. This requires additional assumption on the momentum correlations in the nuclear ground state and on the momentum transfers in the reaction, and involves difficulties of the same nature as when calculating the primary energy distribution.

HARD HEAVY ION REACTIONS

159

Recently we learned about an experiment 12) in which ground state correlations in light Spectator products have been studied in 160 induced reactions on Pb, Au Ni. Results of at3 compatible with the present model are reported, but the authors neglect the decay stage of the reactions.

5. Conclusions Both the analyses and the data are still so premature that one cannot yet conclude that the ground state correlations have been observed. According to the model the correlations are expected to manifest themselves in different ways in the three categories of reaction products: the heavy, the light spectator fragments and the fire ball products. The possibility of observing not too neutron-rich isotopes also exists in hard heavy ion collisions as can be seen from fig. 3, but the model is not too positive on this point. In conclusion one can say that properties of the primary distribution are expected to survive the decay, both in the spectator and in the hot participant material. This gives an interesting possibility of observing isospin ground-state correlations directly in the mass and charge distributions of the reaction products. We want to thank A. Bohr, J. Randrup and W. Swiatecki for helpful discussions. One of us (GF) acknowledges the source of support to both the Danish Research Council and the Commemorative Association of the Japan World Exposition and is indebted to the Niels Bohr Institute for kind hospitality.

References 1) J. Rasmussen, R. Donangeloand L. Oliveira, Proc. of the Institute for Chemicaland PhysicalResearch Symposium on macroscopicfeatures of heavy-ion Collisions and pre-equilibrion processes, Hakone, Japan (1977) p. 440 2) J. Hiifner, K. Sch~ifer and B. Schfirmann, Phys. Rev. C12 (1975) 1888 3) J. D. Bowman, W. J. Swiatecki and C. F. Tsang, LBL-2908(1973); G. D. Westfall et al., Phys. Rev. Lett. 37 (1976) 1202 4) J. Hiifner and J. Knoll, Nucl. Phys. A290 (1977) 460 5) W. D. Myers, Nucl. Phys. A296 (1978) 177 6) W. D. Myers et al., Phys. Rev. C15 (1977) 2032 7) A. Bohr and B. R. Mottelson, Nuclear structure, vol. 2 (Benjamin, NY, 1975) 8) J. P. Bondorf and W. Nfrenberg, Phys. Lett. 44B (1973) 487 9) A. Bohr and B. R. Mottelson, Nuclear structure, vol. 1 (Benjamin, NY, 1969) 10) W. Loveland et al., Phys. Lett. 69B (1977) 284 11) M. Buenerd et al., Phys. Rev. Lett. 37 (1976) 1191 12) D. Scott and P. J. Siemens et al., private communication and to be published