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NUCLEI BREAK IN HIGH ENERGY COLLISIONS.
HOW AND WHY ?
J. Hüfner
To cite this version:
J. Hüfner. NUCLEI BREAK IN HIGH ENERGY COLLISIONS. HOW AND WHY ?. Journal de Physique Colloques, 1986, 47 (C4), pp.C4-3-C4-12. �10.1051/jphyscol:1986401�. �jpa-00225764�
JOURNAL DE PHYSIQUE
Colloque C4, suppl6ment au n o 8, Tome 47, aoGt 1986
NUCLEI BREAK IN HIGH ENERGY COLLISIONS. HOW AND WHY 7
Institut fiir Theoretische Physik, Universitat Heidelberg and
Max-Planck-Institut fiir Kernphysik, 0-6900 Heidelberg, F.R.G.
R'esum'e - C'est une revue sur les r'eactions de fragmentation des noyaux d haute energi'e (au-dessus de 1 GeV/A). Les distributions exp&rimentales en masse, Snergie et angle sont pr'esent'ees pour les fragments. La discussion des modOles thgoriques est limit'ee aux modeles thermiques et ceux de la percolation.
Abstract - The fragmentation of nuclei in high-energy (above 1 GeV/A) proton- nucleus and nucleus-nucleus col 1 isions is discussed. The main experimental facts on the mass, energy and angle distributions of the fragment nuclei are presented. Theoretical descriptions for spa1 lation and mu1 ti fragmentations are 1 imi ted to thermal models and percolation.
I - INTRODUCTION
Heavy-ion collisions in the Fermi energy domain are the subject of the conference.
This is an area in nuclear physics where little is yet known. When exploring new territory one always starts the expedition from famil iar grounds. In our case, the new domain contains heavy-ion reactions at energies around 50 MeV/A. The known re- gions are the low-energy reactions around 10 MeV/A and those at high energies, 1.e.
above 100 MeV/A up to 2 GeV/A. In-my talk, I will restrict myself to the high-energy side and concentrate mainly on the production of heavj fragments. The talk will be essentially an updated version of my recent review /I/.
At relativistic energies the division between artici pants and s ectators has proven very useful. It is based on the following idea-ring b r g y nucleus~
nucleus collision the nucleons move essentially on straight lines in the direction of the beam. The nucleons which are found in the geometrjc overlap between projec- tile and target nuclei interact and are therefore called participants; nucleons out- side the overlap are the spectators (Fig. 1). In the zone of the participants a con- siderable part of the beam energy is converted into heat and high temperatures are reached, while the spectator matter remains relatively cold. Most of the spectacular results at the Bevalac refer to properties of the participants. It all started with a daring hypothesis: the participant nucleons reach thermal equilibrium. This hypo- thesis was put forward by experimentalists to explain their first data. In view of the very short time during which the fireball keeps together, the thermal ization hypothesis met with considerable doubts on the side of the theorists (I was one of them), but we have understood now. Thermal equil i brium then proved extremely fruit- f ul , because many we1 1 -developed concepts from thermodynamics, statistical mechanics and hydrodynamics could be taken over. Temperature, entropy, equation of state and more properties have been established for nuclear matter at high excitation energies and large densities. These properties have already been summarized in two published Physics Reports /2/ and /3/ and two more reviews about hydrodynamics will appear soon.
I shall speak about the properties of heavy fragments where all nuclei with A210 are defined as heavy. These nuclei mainly originate from the spectator remnants of pro- jectile or target, respectively. How do we know? The mean velocity of the observed
Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1986401
JOURNAL DE PHYSIQUE
"gentle"
Fig. 1 - P a r t i c i p a n t s and spectators. A schematic drawing o f a nucleus-nucleus col- l i s i o n i n the impact parameter plane. A p r o j e c t i l e w i t h radius Rp s t r i k e s a t a r g e t w i t h radius R a t impact parameter b. The nucleons i n the overlap region (hatched area) are catfed p a r t i c i p a n t s . The remaining p a r t s o f p r o j e c t i l e and t a r g e t repre- sent the spectator matter. Some o f the p a r t i c i p a n t nucleons (symbolized by the arrows) penetrate i n t o the spectator matter. Large impact parameters lead t o gentle reactions, and v i o l e n t i n t e r a c t i o n s are expected f o r the small values o f b.
fragment i s e i t h e r close t o zero o r close t o the beam v e l o c i t y . There are no heavy fragments w i t h intermediate r a p i d i t y .
Before I enter a discussion o f various experimental and t h e o r e t i c a l r e s u l t s , I want t o present a coarse c l a s s i f i c a t i o n scheme. One b a s i c a l l y distinguishes three mecha- nisms by which heavy fragments can be formed: s a l l a t i o n f i s s i o n and m u l t i f r a men t a t i o n ( t o which degree these mechant isms r e a l k / s ~ ma a t d l ) . T h e a s s i f i c a t i o n scheme i s shown i n Fiq. 2 f o r the fraqmentation o f the t a r q e t
spectator. S p a l l a t i o n leads t o fragments-whose mass i s ciose t o the t a r g e t mais AT and o n l y one heavy fragment i s observed (m=l). F i s s i o n i s defined as the process which leads t o two heavy fragments; t h i s d e f i n i t i o n comprises the well-known thermal
multiplicity
t
mass
= 50 =2AT/3 AT fragment
Fig. 2 - The areas of spallation, f i s s i o n and multifragmentation i n a f i e l d o f mul- t i p l i c i t y o f heavy fragments versus mass o f the fragment.
fission and more violent processes. Reactions in which products with As50 are ob- served are called mu1 ti fragmentation. The prefix "mu1 ti" indicates that several (more than two) fragments are produced. The scheme shown in Fig. 2 mainly applies for A~2200, for smaller nuclei the middle region is absent; spallation dominates together with some mu1 tifragmentation. While spallation and fission are at least partly understood, several models are proposed for multifragmentation.
HOW DO NUCLEI BREAK?
In this section a few basic properties of the fragmentation process are summarized.
I wi 1 1 mainly concentrate on spa1 lation and mu1 ti fragmentation reactions. Further- more I will restrict myself to one-particle inclusive reactions of the form where a projec$ile Ap collides with the target nucleus AT and a fragment nucleus A with momentum p is observed. All information i s contained in the triple differential cross section d3a/d~dS2d~ where $2 and E are the angle and the energy of the fragment.
Mass yield curves
The cross section da/dA=u(A) to observe a particular mass is called the mass ield.
Figure 3 shows two representative results, the mars yield curve for p-Ag and 1 ~ C - l ~ reactions. One clearly observes two distinct domains: in the region of spallation,
&50, the mass yield curve is an exponential, while a power law describes the data for masses where we expect multifragmentation to dominate. Campi et al. /4/ summa-
Fig. 3 - Mass yield for the production o f fragments in reactions with a silver tar- get. The solid lines are empirical fits. The figure is taken from Porile et al. /5/.
rired these findings by a universal dependence of the form
A ,A/AT
u(A)=qr, (F)-Ticrl e
with r=2.5 and ~ ~ 3 . 7 . Note that the results depend on A/AT only,. i.e., the curves scale if plotted in this variable,
C4-6 JOURNAL DE PHYSIQUE
I want t o show another example f o r a power law dependence. I f one p l o t s the m u l t i - fragmentation p a r t o f the cross section on doubly logarithmic paper, one f i n d s a s t r a i g h t 1 in e
I ? a(A)=-r I n A+const. ( 3 )
Figure 4 shows a r e s u l t f o r the c o l l i s i o n " c + A ~ a t 360 MeV p l o t t e d i n t h i s fashion.
Next t o i t i n the same f i g u r e I d i s p l a y the mass d i s t r i b u t i o n o f stone debris a f t e r a c o l l i s i o n where a b a s a l t sphere o f 0.5 cm diameter c o l l i d e s w i t h a b a s a l t sphere o f 5 cm diameter a t a v e l o c i t y o f 2 km/s. I n both experiments one observes a power
Fig. 4 - L e f t hand-side: The charge d i s t r i b u t i o n o f n u c l e i produced i n the frag- mentation o f Ag by a c o l l i s i o n w i t h carbon. The cross sections are p l o t t e d on a doubly logarithmic p l o t . Right hand-side: The mass d i s t r i b u t i o n o f fragments f o r the stone debris observed i n a c o l l i s i o n between two b a s a l t spheres o f 0.5 cm and 5 cm diameter a t the indicated v e l o c i t i e s . The f i g u r e s are taken from a paper by H i f n e r e t a l . /6/.
law f o r the sizes o f fragments w i t h r a t h e r s i m i l a r exponents. Though the two experi- ments d i f f e r by orders o f magnitude i n s i z e and energy, there are also some simi- l a r i t i e s i f scaled i n t h e proper way. I n p a r t i c u l a r , the v e l o c i t y o f sound i n b a s a l t i s 5.5 km/s, so the p r o j e c t i l e v e l o c i t y i n the stone experiment i s about 1/3 o f the v e l o c i t y o f sound. This i s about t h e same r a t i o as f o r the nuclear case, though t h e absolute v e l o c i t y o f sound i n nuclear matter i s much higher (about 0.3 c ) . Do the same experimental r e s u l t s (Fig. 4) imply the same physics? /6/
Momentum d i s t r i b u t i o n s
I pass over t o the d i s t r i b u t i o n d30/dp3 o r equivalently t o d20/dS2d~ and discuss the properties o f fragments which o r i g i n a t e from the decay o f the target. Here the labo- r a t o r y system o f reference i s appropriate. I sumnarize
(1) The angular d i s t r i b u t i o n i s b a s i c a l l y i s o t r o p i c i n a frame which moves forward w i t h a very small v e l o c i t y (a few percent o f the v e l o c i t y of l i g h t ) . This veloc-
ity generally depends on the fragment mass.
(2) The energy spectra show a depression at low energies, which is due to the Coulomb effect and an exponential fa1 1 -off e'E/Eo for energies 1 arge compared to the Coulomb barrier.
(3) The Coulomb barrier is smeared and it is only one half of the size which one would expect for a two-body decay (evaporation of the fragment from the excited target).
(4) The slope parameter Eo is fairly universal Eo-12 MeV, i .e., it only depends in a minor way on the mass A of the fragment, on target,projectile and incident energy. Whether systematic trends (as claimed by Panagiotou /7/) exist and point to a phase transition is not clear to me.
Figure 5 shows a number of distributions in energy and angle for oxygen fragments which are observed in various reactions.
30 60 90 30 60 90
Fragment Energy [MeV]
Fig. 5 - Experimental double differential cross sections for various reactions lead- ing to oxygen. The data are from Warwick et al. /8/, the solid lines are fits by Aichelin et al. / 9 / .
I also want to draw the attention of the audience to measurements of angular distri- butions duldS2 of fragments with mass around 4 0 and I show in Fig. 6 a recent result by Cumming et a1 . /lo/. The angular distribution is forward peaked at projectile energies below 1 GeV and then changes to one peaked at @=go0 with more intensity at 1800 than at 00. This means backward peaking!
I 1 1 - WHY DO NUCLEI BREAK?
This part will be devoted to the theoretical attempts to explain the facts observed for the fragmentation of nuclei in high-energy reactions. I shall briefly touch the spallation reactions and then go into more detail for the multifragmentation process.
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M e V IH
\
Fig. 6 - Comparison of angular d i s t r i b u - tions f o r 3 7 ~ r produced in heavy-ion i n t e r - actions with those f o r similar products i n proton bombardments. Points a r e from the present experiment. Curves without points a r e based on data f o r scandium isotopes (from cumming e t a]. /IO/),
0 5L I I I I I
60 120 I n s
180 slob d*a 1
Spa1 l a t i o n reactions
In a s p a l l a t i o n reaction one fragment i s observed with a mass not too f a r from the t a r g e t mass and a number of small masses, mostly nucleons and a ' s . I t i s generally agreed upon t h a t these products a r i s e in peripheral o r gentle reactions. In the f i r s t s t e p of the reaction only a small amount of the p r o j e c t i l e energy i s deposited i n t o the t a r g e t and a few nucleons a r e ejected from i t . The resulting prefragment then seems t o thermalize and evaporates neutrons, protons and small nuclei u n t i l a s t a b l e fragment i s reached. This explanation f o r the s p a l l a t i o n reactions i s more than 20 years old. Calculations were usually done i n the form of large s c a l e numerical computations using a code f o r an intranuclear cascade followed by an evaporation program. This type of ca7culations can be considerably simplified s o t h a t one i s able t o derive analytical fromulae which e x p l i c i t l y display the exponential be- haviour i n the mass A and the scaling eq. ( 2 ) . Abul-Magd e t a l . /ll/ c a l c u l a t e the f i r s t s t e p i n Glauber theory and derive an e x p l i c i t expression f o r do/dE*, which is the cross section t h a t an amount E* of e x c i t a t i o n energy i s deposited i n t o t h e t a r - get nucleus. Then the mass d i s t r i b u t i o n da/dA i s obtained by assuming t h a t i t takes an average energy E t o remove a mass A A = l . Then
da da *
a = E (E = €(AT - A)). (4) The r e s u l t of such a calculation i s shown i n Fig. 7. The theoretical curves do not contain any f r e e parameter and reproduce the s p a l l a t i o n cross sections i n shape and absolute magnitude. In view of t h i s good agreement I. dare say t h a t s p a l l a t i o n reac- t i o n s can be understood without any sophisticated theory, l i k e those invoked f o r mu1 tifragmentation.
Mu1 t i fragmentation
How does one understand multifragmentation? O u t of the many models I want t o discuss two c l a s s e s of approaches (and the r e l a t i o n between them) f o r which a number of papers were published i n the l a s t few years. Those two c l a s s e s a r e the percolation
Fig. 7 - The mass y i e l d curves f o r t h e fragmentation i n pA c o l l i s i o n s a t 3 GeV (taken from Abul -Magd e t a1 . /11/).
approach and t h e t h e r m a l i t a t i o n hypothesis. Most papers r e s t r i c t themselves t o a d e s c r i p t i o n o f t h e mass d i s t r i b u t i o n s da/dA. As a r a p p o r t e u r I s h a l l a l s o concentrate on t h i s observable and o n l y a t t h e end I w i l l comment on t h e energy d i s t r i b u t i o n s . It i s , I t h i n k , g e n e r a l l y accepted t h a t m u l t i f r a g m e n t a t i o n i s a s t a t i s t i c a l process.
Each i n d i v i d u a l p r o j e c t i le-nucleus s c a t t e r i n g event i s a member o f an ensemble. Any measurement, e.g. t h e d i s t r i b u t i o n da/dA, i s an average over t h e ensemble. Up t o now, complete events ( w i t h o u t neutrons) can o n l y be measured by emulsion techniques, and these emulsion events show how much t h e events d i f f e r f r o m one another. A t h e o r e t i c a l d e s c r i p t i o n s t a r t s from t h e ensemble o f events. I f one i s o n l y i n t e r e s t e d i n t h e mass and charge d i s t r i b u t i o n s , t h e ensemble c o n s i s t s o f a l l p a r t i t i o n s
trhere ai,zi ar$ t h e mass and charge o f one fragment o f t h e p a r t i t i o n . We use $ = (al.. .an) and z = (zl.. .zn). O f course, one has
I: ai = A. and z z i = Z 0 ' ( 6 )
The v a r i o u s t h e o r i e s d i f f e r from each o t h e r by t h e p r o b a b i l i t y W(@) t o f i n d a g i v e n p a r t i t i o n i n t h e ensemble.
I n t h e approach o f minimal i n f o r m a t i o n by A i c h e l i n e t a l . /12/, one assumes t h a t a l l p a r t i t i o n s 6 have equal p r o b a b i l i t y . I n t h i s t h e o r y t h e r e i s no f r e e parameter and one can d e r i v e a simple a n a l y t i c a l formula f o r t h e mass o r charge d i s t r i b u t i o n s . This formula i s amazingly successful.
P e r c o l a t i o n i s t h e mathematical theory o f t h e Swiss cheese. One considers a number o f nucleons d i s t r i b u t e d i n a one, two o r three-dimensional l a t t i c e and u s i n g a random number generator one breaks t h e bonds between some nucleons (bond p e r c o l a t i o n ) o r one removes nucleons from c e r t a i n s i t e s ( s i t e p e r c o l a t i o n ) , Fig. 8. The l o c a t i o n where bonds a r e broken o r nucleons a r e removed i s random, b u t t h e c o n c e n t r a t i o n p of damages i s k e p t t h e same f o r a c e r t a i n c l a s s o f events. I n t h e p a t t e r n o f occu- p i e d and empty s i t e s one recognizes c e r t a i n connected pieces which one i d e n t i f i e s as c l u s t e r s . These c l u s t e r s a r e i d e n t i f i e d as n u c l e a r fragments i f one describes m u l t i -
C4-10 JOURNAL DE PHYSIQUE
fragmentation. I n t h e approaches by Campi e t a l . /13/ and Bauer e t a l . /14/, a l l p a r t i t i o n s Q w i t h t h e same c o n c e n t r a t i o n p o f damages have t h e same p r o b a b i l i t y
W(B) = G(P(@) - P). (7)
I n general t h e p r o b a b i l i t y p i s a d j u s t e d so t h a t t h e mass d i s t r i b u t i o n i s reproduced.
Fig. 8 - Example o f s i t e p e r c o l a t i o n p a t t e r n s on a two-dimensional l a t t i c e w i t h 8 x 8 s i t e s (from a p r e p r i n t by B i r o e t a1 . ) .
F i g u r e 9 from Bauer e t a l . /14/ d i s p l a y s how various q u a n t i t i e s depend on t h e con- c e n t r a t i o n p. One observes a behaviour s i m i l a r t o a phase t r a n s i t i o n a t p=0.75. This t r a n s i t i o n i s v i s i b l e by the sudden drop o f t h e p e r c o l a t i o n p r o b a b i l i t y from one t o zero and by the l a r g e f l u c t u a t i o n a i n t h e m u l t i p l i c i t y d i s t r i b u t i o n . A t t h e same value o f p t h e apparent exponent h has a minimum. T r y i n g t o f i t t h e r e s u l t s o f a p e r c o l a t i o n c a l c u l a t i o n t o experimental fragment d i s t r i b u t i o n s , one o f t e n i s c l o s e t o t h i s k i n d o f phase t r a n s i t i o n . Campi e t a l . /13/ and a l s o Bauer e t a l . /14/ have r e c e n t l y t r i e d t o d e s c r i b e t h e f u l l mass d i s t r i b u t i o n , s p a l l a t i o n and multifragmen- t a t i o n regimes. They i n t r o d u c e a c o n c e n t r a t i o n which depends on t h n m p a c t para- meter p(b). The values o f p a r e small f o r l a r g e impact parameters, corresponding t o p e r i p h e r a l r e a c t i o n s ( s p a l l a t i o n ) and l a r g e f o r t h e c e n t r a l c o l l i s i o n s . This pre- s c r i p t i o n may be f o r m a l i z e d by
The authors succeed i n g i v i n g a u n i f i e d d e s c r i p t i o n o f t h e whole mass d i s t r i b u t i o n . I n t h e thermal approaches, t h e ensemble o f p a r t i t i o n s i s t h e canonical ensemble (Ban-hao /15/ and Bondorf e t a l . /16/) o r t h e microcanonical ensemble (Gross e t a l . /17/). F o r t h e canonical ensemble one f i n d s a p a r t i c u l a r p a r t i t i o n w i t h t h e p r o b a b i l i t y W(B)
where I3 i s t h e i n v e r s e temperature, V t h e a v a i l a b l e volume, E ( B ) t h e energy o f t h i s p a r t i t i o n ( i n c l u d i n g t h e Coulomb energy) and t h e g i n t r a r e t h e degeneracy f u n c t i o n s f o r t h e c l u s t e r s , c a l c u l a t e d from t h e l e v e l d e n s i t i e s . I n those approaches one ad- j u s t s t h e volume V and t h e temperature such as t o f i n d agreement w i t h t h e e x p e r i - mental mass d i s t r i b u t i o n . Here, too, one o f t e n observes v e r y s t r o n g dependencies on these parameters, a phenomenon which smells l i k e a phase t r a n s i t i o n .
Fig. 10 - The r e l a t i v e importance o f v a r i o u s fragmentation modes o f a uZ3' nucleus +
as a f u n c t i o n o f t h e e x c i t a t i o n energy o f t h i s nucleus ( f r o m a microcanonical c a l - c u l a t i o n b y Gross e t a l . /17/).
0.6 0.8
The p r o b a b i l i t y Pperc f o r a percola- t i o n c l u s t e r , i , e . a c l u s t e r which
0.5 extends from one w a l l o f t h e system
t o t h e o p p o s i t e one.
The variance o=J<mL>-<m>~' i n t h e mu1 t i p l i c i t y o f c l u s t e r s .
The apparent exponent A i n t h e mass d i s t r i b u t i o n o f c l u s t e r s o(A) a PTA f o r small A.
F i g . 9 - Various q u a n t i t i e s as a f u n c t i o n o f t h e c o n c e n t r a t i o n o f bonds f o r t h e three-dimensional p e r c o l a t i o n c a l c u l a t i o n by Bauer e t a1 . /14/.
I show i n F i g . 10 a r e s u l t by Gross e t a l . /17/ from a microcanonical c a l c u l a t i o n : the dependence on e x c i t a t i o n energy o f t h e system o f the various fragmentation modes.
100 T 2381)"
!
w B a
"
microcanonical
V, (D
ri -
\ I
\ r
MULTIFRAGMENTATION
E*I M ~ V I
C4-12 JOURNAL DE PHYSIQUE
I V - CONCLUSIONS
M u l t i f r a g m e n t a t i o n o f n u c l e i i s n o t y e t understood. Most t h e o r e t i c a l approaches succeed i n f i t t i n g t h e mass d i s t r i b u t i o n s by a d j u s t i n g few parameters. But t h e r e a r e more data, even f o r s i n g l e p a r t i c l e i n c l u s i v e measurements: t h e d i s t r i b u t i o n s i n an- g l e and energy. I n my opinion, thermal approaches w i l l have a hard time t o d e s c r i b e t h e energy d i s t r i b u t i o n s , because t h e slope o f t h e energy, the apparent temperature, i s v e r y high. I n p e r c o l a t i o n models o r t h e c o l d s h a t t e r i n g o f glass, t h e slope o f the energy d i s t r i b u t i o n i s r e l a t e d t o t h e Fermi energy. Though t h i s f i t s n u m e r i c a l l y amazingly w e l l , I am n o t q u i t e convinced u n t i l I see a k i n d o f microscopic c a l c u l a - t i o n , which shows how t h e Fermi motion o f each c l u s t e r a f t e r p e r c o l a t i o n survives d u r i n g t h e expansion o f the system. F i n a l l y more has t o be done f o r t h e angular d i s t r i b u t i o n s . There a r e q u i t e some s u r p r i s e s . The n e x t years w i l l p r o v i d e us w i t h more data. W i l l they h e l p t o understand t h e physics o r w i l l they even more confuse us? A t l e a s t we l o o k forward t o an e x c i t i n g time.
ACKNOWLEDGEMENTS
I thank J. A i c h e l i n f o r a number o f discussions. This work i s c o n t i n u o u s l y supported by a g r a n t from t h e G e s e l l s c h a f t fiir Schwerionenforschung.
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