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thesis entitled

FIE/4W FEHMWJT ?/ZoDuCTm/U
IN NUCLEA/Z (outflows

presented by

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has been accepted towards fulfillment
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RETURNING MATERIALS:
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remove this checkout from
your record. FINES will
be charged if book is
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stamped below.

 

 

 

 

 

HEAVY FRAGMENT PRODUCTION
IN NUCLEAR COLLISIONS

BY

Jane Ellen Kupstas

A THESIS

Submitted to
Michigan State University
in partial fulfillment of the requirements
for the degree of

MASTER OF SCIENCE

Department of Chemistry

1983

ABSTRACT

HEAVY FRAGMENT PRODUCTION
IN NUCLEAR COLLISIONS

BY

Jane Ellen Kupstas

In the thesis we study the systematics of basic reaction mechanisms
leading to the production of complex fragments, in addition to
considering more exotic processes invoking hydrodynamical instabilities
of nuclear systems. Computer programs were written to calculate the
cross sections of the fragments produced in abrasion and deep inelastic
scattering. These reaction processes occur on different time scales of

22 and 10-21 seconds respectively. The results were

approximately 10-
then compared to experimental data in an attempt to understand the
observed similarity of these superficially different reaction processes.
The calculated cross sections were used to generate the isospin-Z
systematics in agreement with the experimental data.

An "exotic" production mechanism, in which the heated and
compressed nuclear matter is believed to become hydrodynamically
unstable, was also studied. It was found that the systematics were less
well obeyed for these data although the overall trends were rather
similar. Therefore the "exotic" mechanism. if present, is not revealed
through a radical departure from the basic systematics. Although the
present data are inconclusive. there are indications that detailed

studies of proton and heavy ion induced reactions would be useful in

establishing the presence of the "exotic" hydrodynamical instabilities.

ACKNOWLEDGEMENTS

I would like to thank David K. Scott for his patience and guidance
during the last year and a half, without whom this thesis would not be
possible.

I would also like to thank my fiance, David, for his help in
getting this thesis in its final form and for his support during the
writing.

A Lastly, I would like to thank my parents for their love and

support. Without them, this project would never have been accomplished.

ii

TABLE OF CONTENTS

LIST OF FIGURES ..................... .. ............... . .............. iv
1. INTRODUCTIONOOIOOOOOOOIODOIOOOIOOOOOOOCOOOOOOOOO 0000000000 00.0.01
' 1.1 WHY IS FRAGMENT PRODUCTION INTERESTING?....................1
1.2 DESCRIPTION OF REACTION MECHANISMS.......;;................1
1.3 SYSTEMATICS OF COMPLEX FRAGMENT PRODUCTION...... ....... ....5
2. SCHEMATIC MODELS OF HEAVY ION REACTIONS.... .............. ......13
2.1 ABRASION REACTIONS-OOOOOOOOIOOOOOOCOI OOOOOOOOOOOOOOOOOO 0.013
2.2 DEEP INELASTIC TRANSFER REACTIONS.... . ............. . . ..18
3. THEORETICAL INTERPRETATION OF SYSTEMATICS.. ...... .. ......... ...26

A. EXOTIC PRODUCTION MECHANISMS...................................39
‘ ”.1 PHASE TRANSITIONS IN HEAVY ION COLLISIONS.................39
“.2 EQUATION OF STATE.........................................u3

“.3 EXPERIMENTAL EVIDENCE.....................................N7
u.3.1 PROTON INDUCED‘REACTIONS...........................u7

u.3.2 HEAVY ION INDUCED REACTIONS .................... ....53
5. CONCLUSION ......... . ................... ....... ................. 56
APPENDIX A..... ............................................. ........57
APPENDIX s..................... ............... ..... ..... .... ...... ..60
LIST OF REFERENCES .................................................. 6n

iii

LIST OF FIGURES

FIGURE 1: An abrasion reaction where the projectile collides with the
target shearing off a part of both the projectile and the target. The
projectile fragment travels with the incident velocity Binc and the

target fragment travels with the velocity 8. (Here, 8 is the ratio of
the velocity to that of light).(Reference 1) ........................... 2

FIGURE 2: A deep inelastic transfer reaction
(a) the projectile collides with the target
(b) the two stick together and exchange nucleons while rotating

(C) the two come apart both with new masses and N/Z values... ....... ...2

FIGURE 3: A heavy ion collision in which fragmentation is induced by a
mechanical instability

(a) collision

(b) expansion and contraction analogous to a monopole "breathing mode"
(0) fragmentation due to the system's gaining access to the region of

instabilitYOIOOOOOO ...... 00......IO....0......OOOOOOOIO0.0.0.00000000003

FIGURE 4: Qgg-systematics for the system 58Ni + qur (280 MeVJ
(Reference 2)....0... ......... ...OOOOOOOOOOOOOO OOOOOOOOOOOOOOOOO 0.0.0.06
NO

Figure 5: K versus 2 for the Ar projectile on several targets (K is
defined in the text). (Reference 2)....0.0.000.000.0000...0.0.0.0....6

58 . 40

FIGURE 6: Isospin-Q88 systematics for the systems N1 + Ar and

107'109Ag + qur (Reference 2) ........ . ............. . ....... ...........8

iv

FIGURE 7: Isospin-Z systematics for the same systems _as in Figure 6

(Reference 2)....... ...... . ............ ...............................10

FIGUREIB: The functional dependence of the overall gross structure

versus t3 (Reference 2)...............................................1O

12 NO

FIGURE 9: Cross section versus 2 for . C + Ar (top); functional
dependence of linear gross structures versus t3 (bottom).
(Reference 2)....... ...... . ..... ................ ......... .............11

FIGURE 10: 8, v space illustrating the regions in which the four F

functions are valid (Reference 3).....................................15

FIGURE 11: Potential energy landscape; the most probable path of the
system undergoing a deep inelastic transfer reaction, as discussed in
the text, follows the arrow, beginning at the "injection point"

(Reference 8).........................................................19

FIGUREZ122: Theoretically calculated isotope distribution for the

abrasion reaction 1201+ qur (213 MeV/n)..............................26

FIGURE 13: Experimental isotope distribution for the reaction 12C + qur

(213 MeV/n); the theoretical lines are described in reference 13......27

FIGURE 1“: ISOSpin-Z systematics for the reaction 120 + qur (213

Mev/n)................................................................28

FIGURE 15: Functional dependence of gross structures versus t the

3;
slope is O.29........................... ..... .... ...... . ...... ........30

FIGURE 16: Theoretical isotope distribution for the deep inelastic

197Au no

reaction + Ar (21? MeV) ......... .... ............ ..............31

FIGURE 17: Experimental isotope distribution for the reaction 197Au +

qur (217 MeV), theoretical curves are from Reference 9 ............... 32

FIGURE 18: Isospin-Z systematics for the reaction 197Au + qur (217

MeV).......................... ............. ...........................33

FIGURE 19: Functional dependence of gross structure versus t the slope

33
is 0.28 ............................................................... 3’4

FIGURE 20: Contours for the primary product distributions for (a) the
abrasion-ablation model and (c) the Monte Carlo Model and final product
distributions (b) and (d). Notice that the final product contour is
narrower than the contour for the primary products and that the two
final product contours are very similar in spite of the initial

difference in the primaries ........................................... 36

FIGURE 21: A phase diagram for a one component system of normal

matter.. ........ ........................... ...... .............. ....... 39

FIGURE 22: Proposed phase diagram for nuclear matter; normal nuclear

matter corresponds to the point at p/p. - 1 (Reference 16)............A0

FIGURE 23: Isentropes calculated using equation (36) ............ . ..... A3

FIGURE 2“: Energy versus density isentropes, normal nuclear matter

density is 1.0 no where no - p/po (Reference 17) ....... . ...... . ....... A5

FIGURE 25: Production cross section for 21‘Na versus the beam energy for

the reaction of protons + Ag............. ..... ........... ..... ........H7

FIGURE 26: Production cross section for 2”Na versus the beam energy for

the reaction of protons + U.... ........................ ...............A8

vi

FIGURE 27: Production cross section for 2”Na versus the beam energy for

the reaction of 12C + U ............................................... A9

FIGURE 28: Fitting of experimental data (from Reference 18) above the

saturation to the systematics previously discussed... ................. 50

FIGURE 29: The experimental data from Reference 31 describes the
systematics fairly well; the slope is 0.33 ............................ 51

FIGURE 30: Plot from Reference 19 displaying similar behavior to Figures
25, 26 and 27 00000000000000000000 . cccccccccc e ooooooooo no... 0000000 000005”

vii

1 . INTRODUCTION

1.1 WHY IS FRAGMENT PRODUCTION INTERESTING?

The study of the fragments produced in heavy ion collisions is
interesting because it provides one with a means of studying the
evolution of the collision. The nucleus may be shattered into many
fragments or it may be excited and emit fragments on a slower time
scale. By noting the neutron to proton ratio of the emitted fragments
at which the isotope distributions are peaked one can ascertain whether
this number is characteristic of the neutron to proton ratio of the
projectile, the target or of the composite system. This provides one
with information concerning the origin of the fragments produced. Once
the basic mechanisms of fragment production are established through
systematic trends, it may become possible to look for deviations from
the systematics as evidence of more unusual and exotic processes. In
this thesis we develOp basic systematic features of heavy fragment
production and then we attempt to use them as a benchmark in the study

of exotic phenomena.

1.2 DESCRIPTION OF REACTION MECHANISMS
The basic types of reactions with which we will be dealing are the
abrasion reaction and the deep inelastic transfer reaction (DITR). It

will be shown that these two very different types of reactions, which

occur on very different time scales, actually follow similar
systematics. Abrasion reactions (Figure 1) are characteristic of fairly
high energies; they are described by a sudden shearing of the projectile
and target nuclei in the region of overlap. This shearing occurs quite
quickly and thus, no initial excitation energy is injected into either
the target or the projectile. The abraded fragments of the projectile
travel with a velocity very similar to that of the projectile while the
target fragments travel at some very low velocity. Thus, it is possible
in principle, to isolate fragments from either the target or the
projectile. Deep inelastic transfer reactions (Figure 2),, on the other
hand, are characterized by a slower time evolution, when the projectile
and the target stick together and during which various degrees of
freedom, such as the W2 degree of freedom i.e., the ratio of neutrons
to protons, are equilibrated. Although these two mechanisms differ
quite drastically in their basic nature, it hasbeen found that they
lead to fragment production systematics which are very. similar. One
objective of this thesis is to interpret this superficially surprising
similarity by using simple, schematic models of the two processes.
Another type of proposed mechanism - which one might call exotic -
of fragment production in heavy ion collisions is through a mechanical
instability of the compound system (Figure 3). We compare this reaction
process with the DITR and abrasion reactions to establish if the
systematics of fragment production are also obeyed by this mechanism.
What happens here is that the projectile nucleus first comes in and
collides with the target nucleus; the two stick together and form a
compound system in a highly excited state. This compound system may now

oscillate until it reaches a point of mechanical instability, at which

 

FIGURE 1: An abrasion reaction where the projectile collides with the
target shearing off a part of both the projectile and the target. The
projectile fragment travels with the incident velocity sine and the

target fragment travels with the velocity 8. (Here, 8 is the ratio of
the velocity to that of light).(Reference 1).

MSU‘83‘555

_..$_+

(o) (b)

(c)

FIGURE 2: A deep inelastic transfer reaction

(a) the projectile collides with the target

(b) the two stick together and exchange nucleons while rotating
(0) the two come apart both with new masses and N/Z values.

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point it flies apart into a number of fragments. This mechanical
instability is characterized by a negative compressibility, about which

more will be said later.

1.3 SYSTEMATICS OF COMPLEX FRAGMENT PRODUCTION

The systematics which we now describe relate the cross section for
formation of heavy fragments to a simple parameter: e.g., the variation
of the cross section versus Q88 - 6, where Q88 is the reaction Q-value
(111+M2) - (M3+Mu), i.e., the difference in mass between the initial and
the final channel, and O is the pairing energy of the neutrons and the
protons which are transferred; or, the cross section versus the atomic
number of each of the fragments of interest.’ It is interesting to note
that such plots behave very similarly for reactions which take place
through different types of mechanisms. It will also be shown later that
the abrasion mechanism and the deep inelastic transfer mechanism behave
very similarly with respect to a very interesting systematic called the
isospin Z-systematics.

These systematics are presented by Popescu and Popescu2 to describe
the production of fragments in deep inelastic transfer reactions which
employ qur as the projectile. The first is the Q88 systematics in
which the cross sections are plotted versus 088. 6 for the reaction
products (Figure 11). On a logarithmic scale these plots are straight
lines, i.e.:

O - K exp[(Q88 - 6)/T] (1)

where Q8 - (M1+M2) - (M3+Mu), 6 is the pairing energy of the

E
transferred nucleons, T is the temperature parameter of the composite

system and K is a parameter. One uses equation (1) to solve for K since

 

 

 

 

 

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“tudsps awn» FO NC a I
A ‘\ ‘gn ‘ ' ‘ 1
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FIGURE 11: GEE-systematics for the system Ni + Ar (280 MeV)

(Reference 2).

 

 

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Figure 5: K versus 2 fer the qur projectile on several targets (K is
defined in the text). (Reference 2).

everything else in the equation is known either from experimental data
or from a simple calculation.

Popescu and Popescu plotted the parameter K versus Z for several
deep inelastic reactions employing qur as the projectile (Figure 5).
They discovered a surprising result, viz. the values of K for a given
projectile-target system lie on a straight line over as many as 22
orders of magnitude and the characteristics of the lines are found to be
functions of the target used in a particular reaction. This empirical
dependence can be written:

K(Z) a exp(aZ + b) (2)
where a and b are constants for a given reaction. They found that when.
the K values were calculated from the above equation the largest
deviations from the experimental values were smaller than a factor of 2.
Thus, once a and b are found from experimental data in a particular
reaction, one may calculate the cross sections for the unmeasured
isotopes for this deep inelastic reaction to a good approximation.

Another equally striking systematic, is obtained by plotting

C(t3,Q88 - 6) versus Qgg - 6 for each fragment where t - (N-Z)/2 is the

3
isospin of the observed fragment. Figure 6 shows these plots for the

two systems TO7’109Ag + qur (285 MeV) and 58Ni + qur (280 MeV). From

this figure the authors find that at constant isospin values, the
overall variation satisfies a linear dependence (lines are least squares
fits) and at integer t3 values a regular fine structure is superimposed

on the gross structure with maximum amplitudes for t3 - 0 and

monotonically decreasing amplitude with increasing t Lastly, they

3.
find that the slopes of the gross structures decrease monotonically with

increasing t3.

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40

systematics for the systems 58N1 + Ar

Ag + 40A: (Reference 2).

The last systematic presented, of most interest to us, is the

isospin-Z systematics in which one plots O(t 2) versus Z. These plots

3'
are given in Figure 7 for the same systems as above. The authors find
that these systematics exhibit all of the same features as the isospin-

88
structures increase monotonically with increasing t

Q systematics except that in this case, the slopes of the gross
3. The authors then
proceed to extract a dramatic and superficially puzzling result; they

plotted the t values against the slopes of the lines which characterize

3
the gross structures and found a linear dependence with a slope m .
0.29, independent of the target used in the DITR (Figure 8). This
feature led the authors to suggest that the isospin-Z systematics are
determined by the structure of the projectile and not that of the
target. This is a very surprising observation since one would expect
the results of a DITR to be dependent on both the target and the
projectile because of the fact that there is an equilibration of the N/z
ratio between the two.

As with equation (1), the authors write an empirical equation which
yields cross sections in terms of the isospin-Z systematics:

O 0: exp[(mt + n)z] (3)

3
where m and n are constants for a given reaction.
To the authors' surprise, all of the features of the isospin-Z

systematics were found to hold true at high energies for (Figure 9) 12C

+ qur (213 MeV/nucleon) i.e., a total energy of 8520 MeV, some #0 times
higher than in the deep inelastic case. At such an energy, this
reaction is presumed to proceed by the rapid abrasion process mentioned

earlier, which is quite different from deep inelastic scattering. The

trends in the cross section versus Z plots as well as the slope of the

10

 

 

   
 
 

 

 

 

 

 

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FIGURE 7: Isospin-Z systematics for the same systems as in

Figure 6 (Reference 2).

 

 

 

 

 

FIGURE 8: The functional dependence of the overall gross structure

versus t (Reference 2).

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FIGURE 9: Cross section versus 2 for 120 + qur (top); functional

dependence of linear gross structures versus t3 (bottom).

(Reference 2).

12

0 versus isospin line behaved almost identically. In the next chapter:
we will try to show that this similarity can be understood by using
simple schematic models to describe the two reactions which occur in

quite different regimes of energy and time.

2. SCHEMATIC MODELS OF HEAVY ION. REACTIONS

2.1 ABRASION REACTIONS

In.this chapter, we develop simple models to describe fragment
production in both high and low energy reactions. At high energies,
such as the qur reaction at 213 MeV/nucleon described at the end of the
last chapter, a useful approach for the calculation of fragment cross
sections is the abrasion model in which the primary fragments are formed
by the sudden shearing of the projectile nucleus by the target nucleus
without any previous excitation. In this way, the ratio of the protons
to neutrons N/Z, of the primary reaction products is characteristic of
the projectile or of the target rather than a combination of the two.

One modelB’u

which is used to predict the cross sections of the abraded
fragments is the fireball model (see figure 1). This model is based on
the geometry of the reaction and has no adjustable parameters. The

equation used to calculate the cross sections is of the following form:

Z N
O(Z,A) - (29 (n9 O(A) (A)
A
1
a
where:
O(A) - NiEb(a+0.5)]2 - [b(a-0.5)]2} (5)

where A . A1- a; z, n, a refer to the number of nucleons sheared off and

21, A1, N1'refer to the total number of nucleons in the system. This

expression calculates the dispersion in the number of neutrons and

13

1n

protons removed from the projectile or target relative to the number of
ways of distributing neutrons and protons in an assembly of "a"
nucleons. One must first calculate the number of nucleons, a, removed
at impact parameter, b, as follows:

a(v,8) - A1F(v,8) (6)
where F(v,8) is a function of the two dimensionless parameters v and 8.

v specifies the relative sizes of the two nuclei:

v a 1 (7)

 

B - b (8)

 

These two variables range from zero to one and define a square in v.8
space as shown in Figure 10. Gosset et al.3 give the following
approximate formulae for F in each of the four sectors indicated in the
figure:

3/2][1-(B/v)2]1/2

1/2 2 1/2 2 3/2 2 1/2 3
F - 3(1-v). 1‘8 - 1 3(1-v). ' [1-(1-u ) JE1‘(1'H) 1 1'8

Fl - [1-(1-u2)

.8 u U3 _C—
PHI n.3(1-v)1/2(};§)2 - [3(1-v)1/2-1](l:§)3 (9)
U ' v v

FIv ' 1 -

where u - 1/v-1 . R2 /R1.

The four sectors are described by Gosset et al. as corresponding to

the following situations:

"I. A cylindrical hole is gouged in the nucleus A1
(which is larger than A2). '

15

 

 

 

 

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FIGURE 10: 8, v space illustrating the regions in which the four
F functions are valid (Reference 3).

16

II. A cylindrical channel is gouged in A1, 111th a
radius smaller than that of A1. ‘
with 3

III. A cylindrical channel is gouged in A1,

radius larger than that of A1.
IV. All of A1 is obliterated by A
larger than that of A1)."

2 (whose radius is
As can be seen, an explicit function for the evaluation of b is not

written but rather, equation (6) is solved for a number of selected O

values and one then does linear interpolations between these values.
This model does lead to a gaussian-like distribution of isotopes.

However, the width is much greater than the experimental value (as noted

by Morrissey et al.“) because the model assumes that protons and
neutrons can be removed independently. We therefore use a more
realistic fermulation in which correlations of neutrons and protons are
taken into account. We find that the widths of the calculated fragment
distributions then more accurately reflect the widths of the
experimental data. This abrasion-ablation model is described by

Bondorf5

and Scott6, and essentially contains the same basic physics as
the above model.
In order to calculate the cross sections of the primary fragments,

the following equation is employed:

2 2
O m exp - (a 8°) + (t3 t3.) (10)
2 2
20a 20t

3

17

where a=N+Z, the mass number of the primary fragment (the number of

nucleons abraded), the isospin t =(N-Z)/2 and O and O are dispersions

3 3 t3

around the mean values a0 and t3 :
0

Ct - 0.20A:3/3
’ (11)
O - l1.3 at:

and :

(12)

t3o - (No - 2)/2

where Z is the number of protons in the fragment and No is the ideal

number of neutrons in order that the fragment N/Z ratio is equal to the

N/Z of the projectile, i.e., Np/Z - No/Z

P f7

We note that this type of distribution has the potential to
reproduce the type of systematics we described in the previous chapter.
We recall that the constancy of the slope involved two differentiations.
That is to say, one takes the slope of the probability versus atomic
number at constant isospin; these slopes are then plotted against the
isospin and the slope of this plot is calculated, which was found to
have a constant value. Such a constancy can arise from double

differentiation of a quadratic, and we see that the theoretical model

involves such quadratic descriptions.

18

2.2 DEEP INELASTIC TRANSFER REACTIONS

At lower energies the dominant reaction mechanism is thought to be
deep inelastic transfer reactions (DITR) in which the target and
projectile stick together to form a composite system. While they are
stuck together nucleons are constantly being exchanged and the system
strives to reach a minimum in potential energy; however, since the

composite system has a lifetime only of the order of a few 10.22

seconds, full statistical equilibrium cannot, in general, be attained.
That is to say, not all degrees of freedom have enough time to
equilibrate; for example, the mass asymmetry degree of freedom is

usually not relaxed since it takes a time of the order of COMO-22

seconds. The system normally reaches equilibrium with respect to the

neutron to proton degree of freedom (N/Z) since this degree is relaxed

in a time on the order of 10x10-22 secondsg. Upon relaxation of the
neutron to proton degree of freedom the fragments which are formed have

a N/Z ratio equivalent to that of the composite system, e.g., for 197Au

4- qur (total N .1140, total 2 - 97) the value of N/Z - 1110/97 - 1.1111.

In other words, the system strives to obtain the same ratio of protons
to neutrons in the primary fragments as that in the composite system.
An interesting way of looking at this is on a potential energy landscape

(Figure 11). This figure is for the system 56Fe + 136Xe undergoing a'

deep inelastic transfer reaction. As can be seen, the route to the
valley where N/Z equilibration is obtained is quite steep and the

composite system can more or less roll down into the valley on a very

ATOMIC NUMBER 2

19

I l L l l l I

 

60

.351; . .533; ' // . 

. e LAB = 14'

/7 7////
/%%fifit/ESE ‘t/ -

SHELL ’EEF’EQTEVV/
//’E’“MAGIC HOLES”/

   
      
  
  
 

QUANTAL OSCILL.‘
// GIANT MODES? .
/ -
c

      
 

. //

I

I

 

 

/_SHALLOWE
7s 80 ' '

NEUTRON NUMBER N

 

' ' a

5

FIGURE 11: Potential energy landscape; the most probable path of
the system undergoing a deep inelastic transfer reaction, as
discussed in the text, follows the arrow, beginning at the

"injection point" (Reference 8).

20

short time scale as compared to the much less steep route which leads
directly to an equilibration in the mass asymmetry. Thus, although
there are many possible routes (another of which is also illustrated,
where the system overshoots the valley and osCillates from one side to
the other) the composite system prefers to follow the first i.e., down
the steep valley and into N/Z equilibration, then slowly evolves toward
mass symmetry. This preferred path is shown on the figure starting at
the injection point and following the arrow downward. It is thus
obvious that the fragments formed in this type of reaction are created
by a totally different mechanism compared to those formed during the
abrasion reaction.

The reaction model used here for DITR follows the method of Chiang

et al.9. The basic idea is to assume that the N/Z degree of freedom is

much more rapidly relaxed than the mass asymmetry degree i.e., 10::10-22

seconds as Opposed to 60x10.22 seconds. Regardless of- what the mass
asymmetry and neutron to proton asymmetry may be for the initial
channel, the neutron to proton asymetry has enough time to equilibrate
whereas the mass asymmetry does not necessarily have enough time for
equilibration to set in. Chiang et al. give two major assumptions on
which their calculation is based:

"(1) For a given mass ratio in the exit channel, the

N/Z distribution is governed by the potential. energy of

the composite system. If a full statistical equilibrium

is reached for this degree of freedom, then the charge

distribution should obey the Boltzmann equation.

(ii) In order to account for the mass distribution of

the fragments, a classical diffusion model has been
utilized."

21

In the model, the shape of the cOmposite system is approximated by
two spherical liquid drops separated by a distance d. Thus the
potential energy of the system is:

V(Z1,A Z (Z A ) +‘V + V + V (13)

A ) ' VLD(Z1’A1) 1 2' 2 C ROT EC

1' 2' 2 vLD

where Z A A are the charge and mass of each nucleus

1’ 1' 2’ 2

respectivelyy \HJ) is the liquid drop potential of each fragment

10), V is the Coulomb potential

(calculated using Segre's formalism C

between the two drOps and V is the rotational energy of the system:

ROT

2
1 (i + 1)H
VROT f f 2

2uR

(1A)
where, if is the angular momentum in the exit channel, u is the reduced

mass and R is the distance between the two fragments. VEC is the total

potential energy in the entrance channel. (Note that the potential
energy of the system is normalized to zero for the case of the exit
channel and the entrance channel being identical.)

Assuming sticking, i.e. rotation of the rigid composite system, the

orbital angular momentum in the exit channel is found to be:

 

2
<£f> . <21) 2 “82 2 (15)
uR + 2M1R1 + 2M2R2
5 . 5
and R1 and R2 are the radii of each fragment in the Proximity Potential
formalism6’8’9:

R1 . 1.128A1/3(1 - 0.786A12/3) fm (16)

22

The mean value of the entrance channel orbital angular momentum, <21),

is calculated:

£2

2 1/2
(A > - [(2.max + crit)/2] . (17)

i

where zmax and 2c are calculated starting from the equation for the

rit

reaction cross section6:
2 2 2 2 vx
oR - N {ZR - N Rx(1 --E- ) (18)
CM
where x signifies a grazing or critical condition ("grazing" is the

condition where the target and the projectile just touch and "critical"

is the condition below which fusion will occur). Rx, the radii related

to these two conditions, are calculated from:

1/3 1/3
1/3 1/3 (19)
Rgr r,(A1 + A2 ) + d

where r0 - 1.225fm and d - 2fm (d is an adjustable separation

parameter). The center of mass energy is given by:

E A
E _ LAB 2

CM
A1 + A2

(20)

and Vx is the potential at either the grazing or critical distance and

is calculated:

Vx - VN(x) + VC(x) (21)

where VN(x) is the nuclear potential and V is the coulomb potential.

C

The coulomb potential is calculated from:

V .__._ (22)

and the nuclear potential is calculated as follows:

R1R2
vN(x) - unr b¢(E) (23)

R1 I R2

where Y = 0.95/fm2, b - 1 and ¢(E) is:

¢<g < 1.25) - ~1/2(g - 2.511)2 - 0.085(6 - 2.5u)3
¢(£ > 1.25) - ‘3.“37 exp(-5/O.75) (29)
where:
E - s/b = r - (R1 + R2) (25)

using the above expression for the potential energy of the
composite system, the corresponding charge distribution P(Z) for a given
mass asymmetry is a Boltzmann distribution of the form:

'V(Z)/T

P(Z) c e (26)

The nuclear temperature parameter, T, is equal to 2.3 MeV. This value
was calculated from experimental data by Popescu and Popescu. (See

Figure A in which the slope of the production cross section versus Qgg- 6

197

was used in the determination of T.) Note that for the reaction Au +

qur (217 MeV) we expect a value of T - 2.1 MeV from the expression for

a Fermi gas for the compound nucleus, viz.:

E* - A T2 (27)

1—6'

N
where E is the excitation energy of the compound nucleus and A is the

number of nucleons. (We use 16 as the level density parameter since

 

2“

this is the theoretical value found in the Fermi Gas model). The

excitation energy is:

*

E - ECM + Q (28)

equal to 63.8 MeV for this reaction; the center of mass energy is:

_ ELABAZ

A1 + A2

E

CM a 180.” MeV (29)

and Q is the difference in the mass excesses between the initial channel
and the compound nucleus.

Then, the probability for producing the fragment (Z1,A1) is

expressed by:

m1 .111)

P(A ) (30)
ZZP(Z.A1)

FNZ1,A1) - 1

where P(Z1,A1) and ZZP(Z,A1) are obtained using equation (27) and P(A1)
is calculated from a diffusion model:
P(A ) - (unD t )1/2expE-(A - A - v t )2/uD t J A (31)
1 A R 1 P A R A R

where AP is the projectile mass. The reaction time, t is of the order

R.

of 10x1O—22 seconds and can be calculated from the angle of rotation, 6,

the grazing angle, 68, and the angular velocity, w, of the system:

1
tR . a (68 ' 0) (32)

It is a measure of how long the target and projectile nuclei are stuck

together. The drift velocity, VA . 21:10-22 fm/sec and the diffusion

22

coefficient DA - Ax10 fmZ/sec are transport coefficients in the

diffusion model; the values used are those of Chiang et al.

25

Our next objective is to check whether these analytical
descriptions of the two basic reaction mechanisms can successfully

describe the similar systematics discovered in the experimental data.

3. THEORETICAL INTERPRETATION OF THE SYSTEMATICS

To determine whether or not the models for abrasion and deep
inelastic transfer reactions follow the systematics of the experimental
data, computer programs were written to check the theoretical
predictions of the systematics.

First, the theoretical cross section versus atomic number at
constant isospin values were plotted, the slopes of these lines were
then calculated using a least squares fit. These slopes (a) were next
plotted versus the isospin and the slope of this resulting line was
calculated. The important features to compare with the systematics are:
(1) the trend shown in the cross section versus atomic number plots and
(2) the slope of the a versus isospin line. If these cOme out the same
as in the systematics described by Popescu and Popescu, then one may say
that both the DITR and abrasion theories can successfully describe the
systematics in a natural way.

First, for the abrasion reaction, 12C + qur (213 MeV/nucleon),

Figure 12 shows a typical isotope distribution calculated using the
program given in Appendix A. It has a width at one tenth height of 5
mass units which is comparable to the experimental data and peaks at
mass number 18 in agreement with the prediction that the N/Z at the peak
should be equal to the N/Z of the projectile. Figure 13 is the

experimental distribution to be used for comparison to Figure 12.

26

PROBRBILITV

27

RBRRSION: ISOTOPE DISTRIBUTION. Z i 8

 

 

 

 

HESS NUMBER

FIGURE 12: Theoretically calculated isotope distribution for the

abrasion reaction 12C + "oAr (213 Mev/n).

' i
. -1
10' :
10“E :
= 2
I I
10*E :
' 1
10*, :
Z . 1
b , all
1- E‘ .1
‘0‘ 1 1 l l 4 1 J 1 l 4
12 13 I“ 15 16 17 18 19 20 21 22 23

28

 

1 *1‘1 r1111] '"1 *1‘1’1
p-V
,"
l

10

T ITTIIII‘

Cross section (mb)

1
O“ I

T I TTTVITI

 

1 l I L_
14 16 18 20

Fragment mass number

FIGURE 13: Experimental isotope distribution for the reaction 12C + qur
(213 MeV/n); the theoretical lines are described in reference 13.

29

Figure 111, which gives the cross section versus atomic number, follows A
the same trend as the experimental data i.e., the slopes increase with
increasing isospin (compare Figure 7). Now, when these slopes are
plotted versus isospin, Figure 15, we get a very pleasing result; the
slope is equal to 0.29 just as Popescu and Popescu found for the
experimental data. Thus, we may say that our theoretical formulation
for the abrasion reaction follows this systematic.

110 97

Turning then to the DITR, Ar (217 MeV) + 1 Au, Figure 16 gives a
typical isotope distribution calculated using the program given in
Appendix B. It has a width of 6 mass units which agrees well with the
experimental widths. It peaks at mass number 35 which is in general
agreement with the N/Z of the composite system. Figure 17 is an
experimental distribution for comparison. Figure 18 gives the
probability versus atomic number and once again we see the trend of
increasing slope with increasing isospin. Plotting these slopes versus
the isospin (Figure 19) we again obtain a plot with a slope very similar
to that found for the experimental data, viz. a slope of 0.28.

It is quite satisfying to find that the theoretical results follow
the systematics of the experimental data. Although these two types of
reactions are very different in mechanism, the resultant fragment
distributions are rather similar. Basically, both processes lead to
gaussian distributions of cross sections.. The double differentiation
implied by the systematics then leads to the required constancy of
slope. Although the slope is determined by quite different parameters

in these two types of reactions, we have shown that they actually give

rise to the same experimentally determined slope in both cases. The

30

MSU-BS-SGZ

 

U I r

“Ar J'ZIC (ZIE MeV/A)

 

 

 

 

 

 

 

PROBABILITY

 

 

 

FIGURE 111:
(213 MeV/n).

 

Isospin-Z systematics for the reaction 120 +

110

Ar

31

‘ MSU'83'553

 

 

 

 

 

 

I I I r I - l I I
- “Ar + '2 C(213 MeV/A .—
.zo - ' -
.OO
6 - ..
'.20 " "
140- '
-.60 .. .-
’I o I (l) l ITO 1 2 O L

FIGURE 15: Functional dependence of gross structures versus t3; the

slope is 0.29.

PROBRBILITY

32

DITR: ISOTOPE DISTRIBUTION, Z 3 15

 

 

+- I r T r r I l 1 I q
1- .I
1- .1
'- ‘1
.- .1
.. J
1- «I
L d
O!

10 _ _ .
- '1
- -1
- c1
- .‘
1- .1
- «I
" ‘1
h a

' '0

1° - d
b C
D d
I- .1
- d
D I
- Cl
- d
)- c-

10" l l l 1 l l L 4 L

 

 

 

30 31 32 33 .3" .35 38 37 3B 39
HRSS NUMBER

FIGURE 16: Theoretical isotope distribution for the deep
inelastic reaction 197Au + qur (217 MeV).

33

 

P(A) *' [ 2:1511

100 -- - '-

 

 

 

)- 1
IO...’ .1
‘1

 

 

 

 

 

 

OJ L l __D

 

FIGURE 17: Experimental isotOpe distribution for the reaction 197Au +

qur (217 MeV), theoretical curves are from Reference 9.

34

Ila-Br on 197-Ru: PROBRBILITY VS RTONIC O

 

 

 

PROBRBILITY

a
&

 

 

 

 

RTON I C NUMBER

FIGURE 18: Isospin-Z systematics for the reaction 197Au + qur

(217 MeV).

35

DITR: RLPRR vs IsOsPIN

 

1010-

1.00-

RLPNR
a.
a:

l

.50 -
.40 -
.30 -
.20 -

01° *-

 

l

 

 

 

.00 l I L l l 1
.0 .5 1.0 1.5 2.0 2.5 3.0 3.5

ISOSPIN

FIGURE 19: Functional dependence of gross structure versus t

is 0.28.

0.0 “.5

3;

the slope

36

puzzling similarity of the fragment production in two disparate
processes emerges as a coincidence in the combination of various
quantities parameterizing the two reactions.

In this theoretical analysis of the reaction processes, we have
considered only the primary fragment production; secondary emission was
not taken into consideration. Secondary emission is, however, important
because its inclusion will'make the theoretical predictions even more
similar for the two reaction processes. The reasoning is as follows;
the inclusion of secondary emission in the analysis would be expected to
shift the centroid of the isotope distributions by a few mass numbers
through the emission of neutrons and protons; in addition, a slight
narrowing of the isotope distributions would result due to the
constraints imposed by the valley of stability. An example of the
influence of secondary decay is shown in Figure 20; here is plotted the
proton number versus the neutron number at constant cross section values
for the abrasion reaction discussed in this thesis. It can be seen that
the distribution over cross section narrows by approximately four
neutrons when secondary emission is considered. Thus, one would expect
the same type of behavior upon inclusion of secondary emission in our
theoretical models.

Of particular note is the similarity of the two final product
distributions obtained for the two different primary distributions.
(The two cases considered correspond to "no correlations" of the
neutrons and protons and to "correlations" induced by zero point motion
of the giant dipole vibration in the nuclear ground state.) In this
thesis we have discussed primary distributions which are already quite

similar, in spite of their generation by rather different reaction

Proton number

37

 

 

 

 

 

      

 

 

 

 

l I T I
(c) ' ”a (II)
Common (3 '
15—- Primary Wt ‘ ' "
'01
o - d b q
no
0*!
5M- “ ‘ Commin 62¢:le an ‘
'(Inb/ Zuni! IAuniI)
; I l L
' I # I
(d)
Uncorrdsted cascade
'5 _ Final MI: .1
D " "
5- q I- q
l 1 l l l 1 l l
O 5 10 IS 20 O 5 IO 15 20

 

Neutron number

FIGURE 20: Contours for the primary product distributions for (a) the
abrasionsablation mOdel and (c) the Monte Carlo MOdel and final product
distributions (b) and (d). Notice that the final product contour is
narrower than the contour for the primary products and that the two
final product contours are very similar in spite of the initial
difference in the primaries.

38

mechanisms. Although the excitation energies of the primaries in the
two cases will be quite different, the subsequent decay can only enhance
the similarity of between them. For a rough orientation on the primary
excitations, we find for the abrasion reaction that was studied an
excitation of approximately 70 MeV and for the deep inelastic reaction

an excitation of approximately 165 MeV.

11. EXOTIC PRODUCTION MECHANISMS

”.1 PHASE TRANSITIONS IN HEAVY ION COLLISIONS

No study of heavy fragment production is complete without a mention
of exotic production mechanisms which could give rise to deviations from
the systematic trends we have discussed. There has been much discussion
of the "exotic" of late on account of the information such reactions may
yieldon the equation of state of nuclear matter. A knowledge of this
equation of state is quite important if one is to attempt to keep
current with all of the latest speculations on heavy fragment
production. In discussing the equation of state of nuclear matter, the
question which immediately comes to mind is: _why has this study become
interesting? The answer to this question lies within the phase diagram

1'5).

of nuclear matter. The three phases of ordinary matter (Figure 21
and the transitions between them are well known, but there is presently
much controversy over whether or not nuclear matter exhibits any
analogous transformations. It has been suggested by several workers

that nuclear matter may indeed undergo phase transitions under the

conditions which occur during heavy ion collisions. As shown on the

proposed phase diagram of Figure 2216, some quite exotic phases have
been suggested such as quark matter and one analogous to that found in

neutron stars.

39

40

MSU'BB‘SGO

 

 

 

 

 

FIGURE 21: A phase diagram for a one component system of normal
matter.

         

/"

 

 

 

( eeeeeeeeeeeee

70

50

OJ
0

EM (MeV)
25;.

I ISENTROPES

42

MSU-83-561

 

ENERGY vs. DENSITY

   
  

S=4 ,
S=3

OVERSIRESSED-
REGION ,OF REGION

INSTABILITY 1 3:2 _
» I . /S='1.5
3:1

/ /

 

 

 

  

 

 

 

 

 

I. |
0.16 0.25 . 0.35
[0(fm‘3)

FIGURE 23: Isentropes calculated using equation (36).

43

4.2. EQUATION OF STATE

Since compression is expected to occur during high energy heavy ion
collisions one should, in principle, be able to observe the evolution of
both exotic and more normal phases in a collision. In our discussion Of
the basic reaction mechanisms the influence of compression was ignored.
First, let us look at the semi-empirical equation of state for nuclear
matter. It consists of three terms, the ground state energy, the

compressional energy and the thermal energy:
K p _ p 2 "2 p 2/3
_ _ 0
E(T.D) Eo 1' 18( p ) (DE-T 267) (33)
- o

where Eo - -16 MeV, the ground state enerSY: K - 200 MeV, the

3

incompressibility at density po and entropy S . 0; q - 0.16 fm‘ , normal

nuclear density; Er - 38 MeV, the Fermi energy. This equation, as it is

written, could be used to construct isotherms. However, since it is
believed that during nuclear collisions the entropyrather than the
temperature remains constant, this equation must be converted to one
with entropy dependence so that isentropes can be constructed. This can

be done by using the entropy of a system of fermions:

2/3

.2 ‘1
Solving for temperature and squaring:
2
Ne “/3
1 Do

and substituting into the equation of state leads to:

44

’ 2
E(S ) a E + 5.... 11.9.: + 5: 32(1 )2/3 (36)
,p o 18 Do N2 96

This equation can now be used to construct energy versus density
isentropes (Figure 23)._ From these isentropes one can determine the

behavixn~ of a nuclear system as the density is changed from p0. At low

entropy values the energy increases when the density is changed on

either side of po, and at higher entropy values the energy increases

steadily with increasing density.

On these energy versus density isentropes one can identify a region
of "mechanical instability" which is characterized by a negative
incompressibility. This region is non-physical in the sense that as
density increases the pressure of the system decreases. A system which
enters this region will become unstable and undergo fragmentation,
leading to a new mechanism for fragment production, quite different from
those we have discussed earlier. This region is constructed on the
isentrope by first finding the analytical form of the incompressibility
as follows:

k - pegs (37)
39

Thus, it is necessary to find the pressure, P, of the system, viz.:

P =,pz(3£ <38)

89

where E - E(S,p). The analytical form for pressure is:

2e 32
K 3 _ K 2 f 5/3

2
990 900 3“ Do

45

and that for incompressibility is:

2
10:8
300 9pc 27“ Do

This defines a region of very low density compared to normal nuclear
density and therefore it is not possible to reach it directly through' a
heavy ion reaction. It must be reached indirectly. Since nuclear
matter is known to undergo giant monopole oscillations-following a
collision, it is interesting to find the region at densities greater
than normal nuclear density which, if entered through compression during
a heavy ion collision, would allow the system-to oscillate into the
region of instability. This higher density region lies at the same
energy on each "isentrope as the region of instability; thus, access to
this region gives direct access to the region of instability. Since it
has been proposed that this region is accessed during high energy heavy
ion collisions, it is interesting to look at this process on the
isentropes.

Figure 2“ shows isentropes found with a more exact equation of

state?7 than that discussed above; it also shows the unstable and

overstressed regions i.e., the higher density region from which the
system can oscillate into the unstable region. Plotted on the
boundaries of the overstressed and unstable regions are two points A and
8, both possessing the same energy and entropy; if a system is at point
A after a heavy ion collision it will thus be able to move along the
isentrope first decreasing in energy and density and then increasing in
energy and continuing to decrease in density (a giant monopole
oscillation) until reaching point B. At point B the system is in the

unstable zone and will thus undergo fragmentation as shown in Figure 3.

46

.As— monogouonv caxq u 0: anon: o:o.. a" auqmcmu smegma

Landon: daemon .noqonucoau auqncou nsmno> augocm

 

«3N mmDUHm
$5 Emzmo .
QN m._ 0.. A m o
_ . _ .
m
<c\c,
Qw

 

 

 

 

 

 

 

 

 

 

(A3W) AQHBNB 'WNHBLNI

 

47

If this is the case, then one has a route to the overstressed region
which can, hopefully, be used to show whether or not there is some sort

of a phase transition occuring during heavy ion collisions.

“.3 EXPERIMENTAL EVIDENCE

Now, with a little thought, it would appear that less energy should
be required to reach the unstable region when one is considering a
collision induced by a heavy ion than when one is considering a
collision induced by a proton. This is the case because in a heavy ion
collision there will be compression which will move the system into the
higher density region of the isentropes, whereas with a proton there is
essentially no compression and thus the protons must inject a larger
amount of energy to access the unstable region. Essentially in the case
of a proton induced reaction, the system is prepared in a condithmm
vertically above the normal density point on Figure 23.
“.3.1 PROTON INDUCED REACTIONS

Let us now turn to some experimental data in order to determine

whether or not this prediction is upheld. The data which are plotted is
the production cross section of 2“Na for the reaction of protons on

silver (Figure 25), protons on uranium (Figure 26) and 120 on uranium
(Figure 27) all plotted versus beam energy. [Note that these production
cross sections also follow the systematics we have described previously;
Figures 28 and 29 show the cross section versus atomic number and alpha
versus the isospin, respectively, for the reaction of 2h GeV protons on

Cu. The slope of alpha versus isospin shows similar behavdxn~ to those

— '5

p CROSS SECT|0N(mb)

9.

48

 

 

 

 

 

 

MSU’B3’55I
I I ' I T
p + Ag —’ 24N0
I I l L
OJ I ' I0 I00

BEAM ENERGY (GeV)

FIGURE 25: Production cross section for 2"Na versus the beam

energy for the reaction of protons + Ag.

IO

0.!

CROSS SECTION (mb)

0.0!

49

 

 

 

 

 

 

MSU-83-548
I F I l
f 0-0-
I l I I
I00

I I0
BEAM ENERGY (GeV)

FIGURE 26:, Production cross section for 2”Na versus the beam

energy for the reaction of protons + U.

CROSS SECTION (mbI

'5

.0

0.0I

50

MSU‘83‘547

 

1 ' I l
. I2:(:q+.lJi——_gp:Z!4'hd(1

 

 

L ' l l

I

  

I

 

0.: I I0
BEAM E_NERGY(GeV)

FIGURE 27: Production cross section for

energy for the reaction of 720 + U.

IOO

24Na versus the beam

 

CROSS SECTION

51

2“ GeV 9 ON Cu: CROSS SECTION VS RTOI‘IIC #

 

 

 

 

10‘ (x105)
10'
t3=2
3
10' (x IO )
10' t3=372
(xloz)
10' c3=1
($00)
10'
ta= 1’2
10. m
134' 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
10 12 1'4 16 is 20 22 24 26 28 so

R10" 1 C NUI‘IBER

FIGURE 28: Fitting of experimental data (from Reference 18) above the

saturation to the systematics previously discussed.

RLPHR

52

2Q GOV PROTONS ON Cu: RLPHR VS ISOSPIN

 

07° _

.uo "-

010 -

 

 

.00

 

 

 

-.30 l I l l I l
.0 .5 1.0 1.5 2.0 2.5 3.0

ISOSPIN

FIGURE 29: The experimental data from reference 31 describes the
systematics fairly well; the slope is 0.33.

53

of the deep inelastic and abrasion reactions; however, the fact that the
trend is less smooth could be indicative of a new mechanism.]

For protons on silver a saturation of the production cross section
occurs at a beam energy of approximately 2 GeV protons corresponding to
an overall energy of 18.5 MeV/n when all 108 nucleons are considered
i.e., the proton and the 107 nucleons in silver. It is possible that
this saturation is indicative of reaching the region of instability. At
this energy the system breaks up and the injection of additional energy
leads to the same breakup. The data for protons on uranium show a
saturation in cross section at a beam energy of 7.5 GeV protons, this
corresponds to an overall energy of 31 MeV/n for each of the 239
nucleons in the system. This shows that the energy per nucleon
requirement increases with increasing number of nucleons in the system
when a proton beam is used. From Figure 23, this behavior is
anticipated, when one considers that no compression occurs yet many more
nucleons share the excitation in the system. I

I4.3.2 HEAVY ION INDUCED REACTIONS

Next, looking at a heavy ion induced reaction, 12C, incident on a
uranium target, one sees that in terms of the total beam energy,
saturation occurs at approximately the same value as for protons on
uranium, i.e., 7.5 GeV, which corresponds to 30 MeV/n for each of the
250 nucleons in the system. This is to be compared to 31 MeV/n for
protons on uranium. Thus, no definative conclusion can be made
concerning whether or not compression occurs in heavy ion collisions
and, if it does, what its effect is upon the fragmentation of the

system. While researching the literature for these data it became quite

54

evident that there is a great need for more experimentation with heavy
ion beams in the 1-10 GeV/n range in order to pin down, more accurately,
the precise location of the saturation.

Subsequent to the completion of this thesis another group (Alekett

et al.19

) did some work which was presented this summer at the Sixth
High Energy Heavy Ion Study and Second Workshop on Anomalons. They also
plotted the cross section versus projectile energy for protons and for
heavy ions and got the same basic result that we did, although there is

possibly more indication for a difference between the proton and heavy

ion cases. Their data are shown in Figure 30.

55

 

 

 

 

 

 

 

 

102 A I 1 T I I LL“;
101—//’7 f #9 '49—:
f I ‘9 ‘°
ES~1()OI€?//’ "
E, -1 A Heavy Ions + Au -> 24Na _
b 10 I a Heavy Ions + Au -> 46"“380

 

 

P + Au *24Na

 

 

-2_. o _
1O _ o P+'Au #4680
-3? ' l l I l 1 l l l
‘0 o 4 8 12 '16. 20 24 28 1100
I Epr0j(GeV)
XBL 835-931

FIGURE 30: Plot from Reference 19 displaying similar behavior to Figures
25, 26 and 27.

5 . CONCLUSION

In conclusion, I hope to have shown that the study of the
production of heavy fragments is a very interesting and varied field.
The systematics which were presented are only a few of the many proposed
tnxt, they are particularly interesting in demonstrating that two
completely different reaction mechanisms follow the same trends. Our
analysis solves a long-standing puzzle in the literature.

Once such trends are established, deviations may be used to signal
the presence of more unusual, exotic processes. The exotic chapter is
only a tip of the iceberg in that much more has been done, and there is
muCh more to do before the controversy over the existence or non-
existence of instabilities in nuclear systems is decided. We should
note that other types of phase instabilities may also occur, namely a
slower chemical instability in which the heated and compressed nuclear
system may, during the expansion, undergo a phase transition from gas to
liquid. Such a transition could influence the production of complex
fragments, resulting in a distribution different yet again from the
systematics we have described for normal reaction mechanisms as well as
from the distribution characteristic of the faster mechanical

instability.

56

APPENDICES

APPENDIX A

57

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LIST 0? REFERENCES

10.

11.

12.

13.

1H.

15.

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A. Popescu and D.G. Popescu,Phys. Lett. 1018, 15(1981).

J. Gosset, H.H. Gutbrod, W.G. Meyer, A.M. Poskanzer, A. Sandoval,
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D.J. Morrissey, W.R. Marsh, R.J. Otto, w. Loveland and G.T.
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J.P. Bondorf, G. Fai and 0.8. Nielson, Phys. Rev. Lett. 51,
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D.K. Scott, The Current Experimental Situation in Heavy Ion
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V.V. Volkov, Phys. Rep. fig, 93(1978).

D.K. Scott, Extreme States in Nuclear Systems, Invited Papers
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T.H. Chiang, D. Cuerreau, P. Auger, J. Galin, B. Gatty, X. Tarrago,
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J.R. Birkelund and J.R. Huizenga, Phys. Rev. C17, 126(1978).

J. Blocki, J. Randrup, W.J. Swiatecki and C.F. Tsang, Ann. Phys.
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Y.P. Viyogi, T.J.M. Symons, P. Doll, D.E. Greiner, H.H. Heckman,
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D.J. Morrissey, L.F. Oliveira, J.O. Rasmussen, G.T. Seaborg, Y.
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G.W. Castellan, Physical Chemistry (Addison-Wesley Publishing
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54

16.

17.

18.

19.

20.
21.
22.

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