aa r X i v : . [ nu c l - t h ] S e p P RAMANA c (cid:13) Indian Academy of Sciences— journal ofphysics
Nuclear Multifragmentation: Basic Concepts
G. CHAUDHURI a , ∗ , S. MALLIK a , † , S. DAS GUPTA b , ‡ a Physics Group, Variable Energy Cyclotron Centre, 1/AF Bidhannagar, Kolkata 700064, India b Physics Department, McGill University, Montr´eal, Canada H3A 2T8
Abstract.
We present a brief overview of nuclear multifragmentation reaction. Basic formalismof canonical thermodynamical model based on equilibrium statistical mechanics is described. Thismodel is used to calculate basic observables of nuclear multifragmentation like mass distribution,fragment multiplicity, isotopic distribution and isoscaling. Extension of canonical thermodynamicalmodel to a projectile fragmentation model is outlined. Application of the projectile fragmentationmodel for calculating average number of intermediate mass fragments and the average size of largestcluster at different Z bound , differential charge distribution and cross-section of neutron rich nucleiof different projectile fragmentation reactions at different energies are described. Application ofnuclear multifragmentation reaction in basic research as well as in other domains is outlined. Keywords.
Multifragmentation, canonical thermodynamical model, projectile fragmentation
PACS Nos.
1. Introduction
The study of nuclear multifragmentation [1–6] is an important technique for understand-ing the reaction mechanism in heavy ion collisions at intermediate and high energies.Due to collision of projectile and target nuclei, an excited nuclear system is formed. Ifits excitation energy is greater than a few MeV/nucleon, then it breaks into many nuclearfragments of different masses. This is known as nuclear multifragmentation. Here ’multi’indicates ’more than two’. Generally in nuclear fission process the compound nucleusbreaks into two fission fragments. Therefore multifragmentation can be considered asthe higher energy version of fission. Usually in nuclear multifragmentation reactions, re-quired energy of the projectile beam produced from particle accelerator varies from fewMeV/nucleon to few GeV/nucleon. The time scales involved in nuclear multifragmenta-tion reaction are at most of the order of several hundred fm/c ( fm/c= . x − sec).Different theoretical models have been developed for throwing light on the nuclear mul-tifragmentation reaction and for explaining the relevant experimental data. The theoreticalmodels can be classified into two main categories: (i) Dynamical models (Boltzmann-Uehling-Uhlenbeck (BUU) model [1], Antisymmetrised Molecular Dynamics (AMD)model [2], Isospin dependent quantum molecular dynamics (IQMD) model [3] etc.) and ∗ [email protected] † [email protected] ‡ [email protected]
2. Statistical Models of Multifragmentation
Nuclear multifragmentation reactions are successfully described by statistical modelsbased on equilibrium scenario of different excited fragments at freeze-out condition [4–6]. In statistical models, one assumes that depending upon the original beam energy, thedisintegrating system may undergo an initial compression and then begins to decompress.As the density of the system decreases,higher density regions will develop into com-posites. As this collection of nucleons begins to move outward, rearrangements, masstransfers, nuclear coalescence and most physics will happen until the density decreasesso much that the mean free paths for such processes become larger than the dimension ofthe system.This condition is termed as freeze-out [5].The disintegration of excited nuclei can be studied by implementation of different sta-tistical ensembles. Calculation by microcanonical ensemble is most realistic but verydifficult to implement. Usually the grand canonical models are easily solved and they aremore commonly used. In grand canonical models total mass or total charge fluctuation isallowed but physically it is not allowed in intermediate energy nuclear reactions. Statis-tical multifragmentation model of Copenhagen [5], the microcanonical models of Gross[6] and Randrup and Koonin [7] are commonly used. Canonical Thermodynamical Model(CTM)[4] was introduced later is easier to implement analytically and its main advantageis that one can eliminate the computationally intensive Monte Carlo procedures by usingthe recursive technique of Chase and Mekzian [8]. The results from the models based ondifferent ensembles converge only under certain conditions for finite nuclei [9, 10].
3. Canonical thermodynamical model (CTM)
Assuming that a system with A nucleons and Z protons at temperature T , has expandedto a higher than normal volume, the partitioning into different composites can be calcu-lated according to the rules of equilibrium statistical mechanics. In a canonical model, thepartitioning is done such that all partitions have the correct A , Z (equivalently N , Z ).2he canonical partition function is given by Q N ,Z = X Y ω n I,J
I,J n I,J ! (1)Here the sum is over all possible channels of break-up (the number of such channelsis enormous); ω I,J is the partition function of one composite with neutron number I and proton number J respectively and n I,J is the number of this composite in the givenchannel. The one-body partition function ω I,J is a product of two parts: one arising fromthe translational motion and another is the intrinsic partition function of the composite: ω I,J = Vh (2 πmT ) / A / × z I,J ( int ) (2)Here V is the volume available for translational motion; V will be less than V f , the volumeto which the system has expanded at break up. We use V = V f − V , where V is thenormal nuclear volume. For all calculations in section 4 we have considered V f = 6 V ,which is obtained from experimental measurements and theoretical data fitting.The average number of composites with I neutrons and J protons can be written as h n I,J i = ω I,J Q N − I,Z − J Q N ,Z (3)There are two constraints: N = P I × n I,J and Z = P J × n I,J . Substituting eq.(3)in these two constraint conditions, two recursion relations [8] can be obtained. Any onerecursion relation can be used for calculating Q N ,Z . For example Q N ,Z = 1 N X I,J Iω I,J Q N − I,Z − J (4)We list now the properties of the composites used in this work. The proton and the neu-tron are fundamental building blocks thus z , ( int ) = z , ( int ) = 2 where 2 takescare of the spin degeneracy. For deuteron, triton, He and He we use z I,J ( int ) =(2 s I,J + 1) exp( − βE I,J ( gr )) where β = 1 /T, E I,J ( gr ) is the ground state energy ofthe composite and (2 s I,J + 1) is the experimental spin degeneracy of the ground state.Excited states for these very low mass nuclei are not included. For mass number A = 5 and greater we use the liquid-drop formula. For nuclei in isolation, this reads ( A = I + J ) z I,J ( int ) = exp 1 T [ W A − σ ( T ) A / − κ J A / − C s ( I − J ) A + T Aǫ ] (5)The expression includes the volume energy, the temperature dependent surface energy,the Coulomb energy and the symmetry energy. The term T Aǫ represents contributionfrom excited states since the composites are at a non-zero temperature.We also have to state which nuclei are included in computing Q N ,Z (eq.(4)). For I, J , we include a ridge along the line of stability. The liquid-drop formula above alsogives neutron and proton drip lines and the results shown here include all nuclei withinthe boundaries. The long range Coulomb interaction between different composites isincluded by the Wigner-Seitz approximation[5].
4. Results from CTM
Important properties of nuclear multifragmentation like mass distribution, fragment mul-tiplicity, isotopic distribution and isoscaling are studied theoretically by using canonicalthermodynamical model (CTM). 3.1
Mass Distribution -5 -3 -1 Y ( A ) A Σ Y ( A ) T (MeV)
Figure 1. (Color online) Left panel: Theoretical mass distribution from A = 168 and Z = 75 system studied at T=3 MeV (blue dotted line), 5 MeV (red dashed line) and 7MeV (green solid line). Right panel: Variation of total multiplicity with temperature. Mass distribution of different fragments produced from the system of mass A = 168 and charge Z = 75 (it represents Sn + Sn central collisions after preequilibriumparticle emmision), is calculated at three different temperatures and is shown in left panelof Fig. 1. At T =3.0 MeV (lower excitation of compound nuclear system) fission is thedominating channel i.e. the multiplicity (total number of fragments) is about 2. Butat T =5 MeV (moderate excitation), fission channel disappears and multi-fragmentation(breaking into large number of fragments) is the dominant process with a large number ofintermediate mass fragments being formed. With further increase of temperature from 5MeV to 7 MeV (very high excitation) the system mainly breaks into a larger number ofsmaller mass fragments. The variation of total fragment multiplicity with temperature isshown in the right panel of Fig. 1.4.2 Isotopic Distribution
Isotopic distribution of Z = 8 and fragments produced by multifragmentation of A =168 and Z = 75 at two different temperatures T = Z = 8 and produced from two differentsources of charge Z = 75 and masses A = 168 , A = 186 is plotted in Fig. 2(c)and 2(d). From the isotopic distributions it is clear that the production of neutron richfragments are more from the neutron rich source Z = 75 , A = 186 compared to theother less neutron-rich Z = 75 , A = 168 .4.3 Isoscaling
Isoscaling [11, 12] is an important property for studying the symmetry energy in interme-diate energy nuclear reactions . It is observed both theoretically and experimentally thatthe ratio of yields R = Y ( N, Z ) /Y ( N, Z ) from two reactions and having differ-ent isospin asymmetry ( is more neutron rich than ) exhibit an exponential relationshipas a function of neutron(N) and proton(Z) number i.e. R = Y ( N, Z ) /Y ( N, Z ) = C exp( αN + βZ ) (6)4
10 15 2010 -4 -3 -2 -1 Z=8 (a)
10 15 20 25 3010 -8 -6 -4 -2 Z=14 (b) -4 -3 -2 -1 Z=8 (c)
10 15 20 25 3010 -8 -6 -4 -2 Z=14 (d) Y ( N , Z ) N Figure 2. (Color online) Upper Panels: Theoretical isotopic distribution from A = 168 , Z = 75 calculated at T=5 MeV (red dashed lines) and 7 MeV (greensolid lines). Lower panels: Theoretical isotopic distribution from A = 168 , Z = 75 (red dashed lines) and A = 186 , Z = 75 (black dotted lines) calculated at T=5 MeV. where α and β are isoscaling parameters and C is a normalization constant.To study the isoscaling in nuclear multifragmentation, we take the dissociating systemshaving same Z = Z = 75 but A = 168 and A = 186 . The ratio R is plotted inFig. 3(a) as function of the neutron number for Z = 6 , , and at T = 5 MeV. It isseen that the fragments produced by CTM exhibit very well the linear isoscaling behavior.The variation of the isoscaling parameter α with temperature in Fig. 3(b) shows that α gradually decreases with T . α is related to the symmetry energy coefficient C s used inthe liquid-drop formula in eq.(5). -4 -2 Z=12Z=10Z=8
Z=6 R ( N , Z ) T (MeV) N α Figure 3. (Color online) Left panel:Ratios( R ) of multiplicities of the fragments ( N, Z ) where reaction 1 is A = 168 , Z = 75 and reaction 2 is A = 186 , Z = 75 .Right panel: Variation of isoscaling parameter ( α ) with temperature. . Extension of CTM to a model for Projectile Fragmentation Projectile fragmentation is a a very useful technique for the production of radioactiveion beam and is also important for astrophysical research. This led to the extension ofthe canonical thermodynamical model and subsequently development into a model forprojectile fragmentation [13–15].The model for projectile fragmentation reaction consists of three stages: (i) abrasion,(ii) multifragmentation and (iii) evaporation. In heavy ion collision, if the beam energyis high enough, then in the abrasion stage at a particular impact parameter three differentregions are formed: (i) projectile spectator or projectile like fragment (PLF) moving inthe lab with roughly the velocity of the beam, (ii) participant which suffer direct violentcollisions and (iii) target spectator or target like fragment (TLF) which have low velocitiesin the laboratory. Here we are interested in the fragmentation of the PLF. Using straight-line geometry average number of protons and neutrons present in the projectile spectatorat different impact parameters are calculated. The total cross-section of abraded nucleushaving Z s protons and N s neutrons is [14, 15] σ a,N s ,Z s = X i σ a,N s ,Z s ,T i (7)where the sum is done over all impact parameter intervals and σ a,N s ,Z s ,T i = 2 π h b i i ∆ bP N s ,Z s ( h b i i ) (8)where P N s ,Z s ( h b i i ) is the probability of formation of a projectile spectator having Z s protons and N s neutrons obtained by using the minimal distribution within the impactparameter interval ∆ b around h b i i [13].The multifragmentation stage calculation of each PLF created after abrasion at dif-ferent impact parameters is done separately by using the Canonical ThermodynamicalModel described in section 3. The impact parameter dependence of freeze-out tempera-ture is considered as T ( b ) = 7 . − . A s ( b ) /A ) [15] where A s ( b ) is the mass of theprojectile spectator created at impact parameter b and A is the mass number of originalprojectile. So freeze-out temperature of the projectile spectator is independent of the in-cident beam energy but it depends on the wound in the projectile. This parametrizationof temperature profile is obtained by looking at many pieces of data from many nuclearreactions. Almost same PLF size and similar trend of temperature profile is obtained frommicrooscopic calculations [16, 17] also. The freeze-out volume in multifragmentation is V f ( b ) = 3 V ( b ) where V ( b ) is the volume of projectile spectator created at b . Using CTMfor an abraded system N s , Z s at temperature T i average population of the composite withneutron number n , proton number z is calculated in the multifragmentation stage. Denot-ing this by M N s ,Z s ,T i n,z and summing over all the abraded N s , Z s that can yield n, z , theprimary cross-section for n, z is σ prn,z = X N s ,Z s ,T i M N s ,Z s ,T i n,z σ a,N s ,Z s ,T i (9)The excited fragments produced after multifragmentation decay to their stable groundstates. Its can γ -decay to shed its energy but may also decay by light particle emission tolower mass nuclei. We include emissions of n, p, d, t, He and He. Particle decay widthsare obtained using the Weisskopf’s evaporation theory [18]. Fission is also included as ade-excitation channel though for the nuclei of mass <
100 its role will be quite insignifi-cant. Details of the implementation of evaporation model can be found in [19]6 . Results from Projectile Fragmentation Reactions
The projectile fragmentation model is used to calculate the basic observables of projectilefragmentation like the average number of intermediate mass fragments ( M IMF ), the aver-age size of the largest cluster and their variation with bound charge ( Z bound ), differentialcharge distribution,cross-section of neutron rich fragments for different nuclear reactionsat intermediate energies with different projectile target combinations.6.1 M IMF variation with Z bound
10 20 30 400.00.51.01.52.02.5 (a) M I M F Sn+ Sn (b) Sn+ Sn Z bound Figure 4. (Color online) Mean multiplicity of intermediate-mass fragments M IMF , asa function of Z bound for (a) Sn on
Sn and (b)
Sn on
Sn reaction obtainedfrom projectile fragmentation model (red solid lines). The experimental results areshown by the black dashed lines.
The variation of the average number of intermediate mass fragments M IMF ( ≤ Z ≤ )with Z bound (= Z s minus charges of all composites with charge Z = 1 ) for Sn on
Snand
Sn on
Sn reactions is shown in Fig.4. The theoretical calculation reproduces theaverage trend of the experimental data very well. The experiments are done by ALADINcollaboration in GSI at 600A MeV [20]. At small impact parameters, the size of theprojectile spectator (also Z bound ) is small and the temperature of the dissociating systemis very high. Therefore the PLF will break into fragments of small charges (mainly Z =1 , ). Therefore the IMF production is less. But at mid-central collisions PLF’s are largerin size and the temperature is smaller compared to the previous case, therefore largernumber of IMF’s are produced. With further increase of impact parameter, though thePLF size (also Z bound ) increases, the temperature is low, hence breaking of dissociatingsystem is very less (large fragment remains) and therefore IMF production is less.6.2 Differential charge distribution
The differential charge distributions for different intervals of Z bound /Z are calculated bythe projectile fragmentation model for Sn and
Sn on
Sn reactions and comparedwith experimental data [20]. This is shown in Fig.5. For the sake of clarity the distri-butions are normalized with different multiplicative factors. At peripheral collisions (i.e. . ≤ Z bound /Z ≤ . ) due to small temperature of PLF, it breaks into one large fragmentand small number of light fragments, hence the charge distribution shows U type nature.7
10 20 30 4010 -6 -3 -6 -3 (a) x10 x10 x10 x10 -2 x10 -4 z bound /z =0.0-0.2z bound /z =0.2-0.4z bound /z =0.4-0.6z bound /z =0.6-0.8z bound /z =0.8-1.0 Proton Number(Z) C r o ss - sec t i on ( m b ) Sn+ Sn Sn+ Sn (b) z bound /z =0.6-0.8z bound /z =0.0-0.2z bound /z =0.2-0.4z bound /z =0.4-0.6z bound /z =0.8-1.0x10 x10 x10 x10 -2 x10 -4 Figure 5. (Color online) Theoretical differential charge cross-section distribution (redsolid lines) for (a)
Sn on
Sn and (b)
Sn on
Sn reaction compared with theexperimental data (black dashed lines).
But with the decrease of impact parameter the temperature increases, the PLF breaks intolarger number of fragments and the charge distributions become steeper. The features ofthe data are nicely reproduced by the model.6.3
Size of largest cluster and its variation with Z bound (a) Sn+ Sn (b) Sn+ Sn Z m ax / Z Z bound /Z Figure 6. (Color online) Z max /Z as a function of Z bound /Z for (a) Sn on
Snand (b)
Sn on
Sn reaction obtained from projectile fragmentation model (red solidlines). The experimental results are shown by the black dashed lines.
Average size of the largest cluster produced at different Z bound values is calculated inthe framework of projectile fragmentation model for Sn and
Sn on
Sn reactions.In Fig.6 the variation of Z max /Z ( Z max is the average number of proton content in thelargest cluster) with Z bound /Z obtained from theoretical calculation and experimentalresult are shown. Very nice agreement with experimental data is observed.8.4 Cross-section and binding energy of neutron rich nuclei
Projectile fragmentation cross-sections of many neutron-rich isotopes have been mea-sured experimentally from the Ca and N i beams at
MeV per nucleon on Be and T a targets [21]. Our theoretical model reproduces the cross-sections of projectile frag-mentation experiments very well [13–15]. A remarkable feature is the co-relation betweenthe measured fragment cross-section ( σ ) and the binding energy per nucleon( B/A ). Thisobservation has prompted attempts of parametrization of cross-sections [22–24]. Onevery successful parametrization is σ = Cexp [ BA τ ] (10)Here τ is a fitting parameter. In this parametrization we have not considered the pairing
28 32 36 408.08.28.4 B / A ( M e V ) Mass Number (A) A P -5 -3 -1 C r o ss - sec t i on ( m b ) Figure 7.
Fragment cross-section (circles joined by red dotted line) for Ni on Be reaction and binding energy per nucleon (squares joined by black solid line) plotted asa mass number for Z = 15 isotopes. energy contribution in nuclear binding energy. Here we have calculated production cross-sections Z = 15 isotopes for N i on Be reaction from projectile fragmentation modeland plotted in log scale in Fig.-7 (circles joined by red dotted line). The variation of thetheoretical binding energy per nucleon for same isotopes of Z = 15 in linear scale is alsoshown in the same figure (squares joined by black solid line). The similar trend of thecross-section curve (in log scale) and binding energy curve (in linear scale) confirms thevalidity of above parametrization from our model. By this method we can interpolate (orextrapolate) the cross-section of an isotope if the binding energy is known. We can alsoestimate the binding energy of an isotope by measuring its cross-section experimentally..
7. Application of Nuclear Multifragmentation
Nuclear multifragmentation is very useful for studying nuclear liquid gas phase transitionand for investigation of nuclear matter at sub-saturation densities. Projectile fragmenta-tion is very useful technique for production of radioactive ion beam and is useful for nu-clear structure studies as well as for astrophysical research. Nuclear multifragmentationcan be used for spallation reaction (nuclear power production), nuclear waste manage-ment (environment protection), proton and ion therapy (medical applications), radiation9rotection of space missions (space research) etc. Thus nuclear multifragmentation is animportant tool in basic research as well as in a wide variety of other applications.
8. Summary
The study of nuclear multifragmentation is an important area of research in intermediateenergy heavy-ion collisions. The canonical thermodynamical model which is based on an-alytic evaluation of the partition function has been used to calculate different observablescharacterizing the multifragmentation reaction and some simple results are displayed.This simple analytical model is extended to develop a model for projectile fragmenta-tion which is important for the study of exotic nuclei as well as for astrophyical research.A few typical observables like average multiplicity of intermediate mass fragments, dif-ferential charge distribution, cross-section of neutron rich nuclei, size of the largest clusterand its variation with Z bound are calculated using this model and their good agreementwith experimental data confirms the justification of the assumptions made in the model.Apart from basic research, the disintegration of the excited nuclei into many pieces alsofinds its application in a wide variety of other fields. References [1] G. F. Bertsch and S. Das Gupta, Phys. Rep , 189 (1988).[2] A. Ono and H. Horiuchi, Prog. Part. Nucl. Phys , 501 (2004).[3] C. Hartnack et al., Eur. phys. J. A
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