Improvements to model of projectile fragmentation
aa r X i v : . [ nu c l - t h ] A ug Improvements to model of projectile fragmentation
S. Mallik, G. Chaudhuri
Variable Energy Cyclotron Centre, 1/AF Bidhannagar, Kolkata 700064, India
S. Das Gupta
Physics Department, McGill University, Montr´eal, Canada H3A 2T8 (Dated: October 7, 2018)
Abstract
In a recent paper [1] we proposed a model for calculating cross-sections of various reaction prod-ucts which arise from disintegration of projectile like fragment resulting from heavy ion collisionsat intermediate or higher energy. The model has three parts: (1) abrasion, (2) disintegration of thehot abraded projectile like fragment (PLF) into nucleons and primary composites using a model ofequilibrium statistical mechanics and (3) possible evaporation of hot primary composites. It wasassumed that the PLF resulting from abrasion has one temperature T . Data suggested that whilejust one value of T seemed adequate for most cross-sections calculations, it failed when dealingwith very peripheral collisions. We have now introduced a variable T = T ( b ) where b is the impactparameter of the collision. We argue there are data which not only show that T must be a functionof b but, in addition, also point to an approximate value of T for a given b . We propose a verysimple formula: T ( b ) = D + D ( A s ( b ) /A ) where A s ( b ) is the mass of the abraded PLF and A isthe mass of the projectile; D and D are constants. Using this model we compute cross-sectionsfor several collisions and compare with data. PACS numbers: 25.70Mn, 25.70Pq . INTRODUCTION In a recent paper [1] we proposed a model of projectile mutifragmentation which was ap-plied to collisions of Ni on Be and Ta at 140 MeV/nucleon and Xe on Al at 790 MeV/nucleon.The model gave reasonable answers for most of the cross-sections studied. The model re-quires integration over impact parameter. For a given impact parameter, the part of the pro-jectile that does not directly overlap with the target is sheared off and defines the projectilelike fragment(PLF). This is abrasion and appealing to the high enrgy of the beam, is calcu-lated using straight line geometry. The PLF has N s neutrons, Z s protons and A s (= N s + Z s )nucleons (the corresponding quantities for the full projectile are labelled N , Z and A ).The abraded system N s , Z s has a temperture. In the second stage this hot PLF expands toone-third of the normal nuclear density. Assuming statistical equilibrium the break up of thePLF at a temperature T is now calculated using the canonical thermodynamic model(CTM).The composites that result from this break up have the same temperature T and can evolvefurther by sequential decay(evaporation). This is computed. Cross-sections can now becompared with experiment. The agreements were reasonable except for very peripheral col-lisions and it was conjectured in [1] that the main reason for this discrepancy was due tothe assumption of constant T over all impact parameters.Full details are provided in [1]. Our aim here is to improve the model by incorporatingan impact parameter dependence of T = T ( b ). While we were led to this by computing thecross-sections of very large PLF’s (which can only result from very peripheral collisions),the effect of temperature dependence is accentuated in other experiments. In fact theseexperiments can be used, with some aid from reasonable models, to extract “experimental”values for temperature T at each b . We spend considerable time studying this although ourprimary aim was and is the computation of cross-sections from a theoretical model. II. BASICS OF THE MODEL
Consider the abrasion stage. The projectile hits the target. Use straight line geometry.We can then calculate the volume of the projectile that goes into the participant region(eqs.A.4.4 and A.4.5 of ref [2]). What remains in the PLF is V . This is a function of b . Ifthe original volume of the projectile is V , the original number of neutrons is N and the2riginal number of protons is Z , then the average of neutrons in the PLF is < N s ( b ) > =[ V ( b ) /V ] N and the average number of protons is < Z s ( b ) > = [ V ( b ) /V ] Z ; < N s ( b ) > (andsimilarly < Z s ( b ) > ) is usually a non-integer. Since in any event only an integral number ofneutrons (and protons) can appear in a PLF we need a prescription to get integral numbers.Let the two nearest integers to < N s ( b ) > be N mins ( b ) and N maxs ( b ) = N mins ( b ) + 1. Weassume that P N s ( b )=the probability that the abraded system has N s neutrons is zero unless N s ( b )is either N mins ( b ) or N maxs ( b ). Let < N s ( b ) > = N mins ( b ) + α where α is less than 1.Then P ( N maxs ( b )) = α and P ( N mins ( b ) = 1 − α . Similar condions apply to P Z s ( b ). Theprobability that a PLF with N s neutrons and Z s protons materializes from a collision atimpact parameter b is given by P N s ,Z s ( b ) = P N s ( b ) P Z s ( b ). Once this PLF is formed it willexpand and break up into composites at a temperature T . We use CTM to obtain these.All the relevant details of CTM can be found in [1] and [3]. We will not repeat these here.There can be very light fragments, intermediate mass fragments (defined more preciselyin the next section) and heavier fragments. As the fragments are at temperature T it ispossible some of these will sequentially decay thereby changing the final population whichis measured experimentally. Details of evaporation can be found in [1] and [4]. III. ARGUMENTS FOR b -DEPENDENCE OF TEMPERATURE Experimental data on M IMF as a function of Z bound (see Fig.1 in [5]) probably provide thestrongest arguments for needing an impact parameter dependence of the temperature. Here M IMF is the average multiplicity of intermediate mass fragments (in this work those with z between 3 and 20) and Z bound =sum of all charges coming from PLF minus particles with z =1. For ease of arguments we will neglect, in this section, the difference between Z bound and Z s , the total charge of all particles which originate from the PLF. A large value of Z bound (close to Z of the projectile) signifies that the PLF is large and the collision is peripheral(large b ) whereas a relatively smaller value of Z bound will imply more central collision (small b ). For equal mass collision Z bound goes from zero to Z , the total charge of the projectile.The following gross features of heavy ion collisions at intermediate energy are known. Ifthe excitation energy (or the temperature) of the dissociating system is low then one largefragment and a small number of very light fragments emerge. The average multiplicity ofIMF is very small. As the temperature increases, very light as well as intermediate mass3ragments appear at the expense of the heavy fragment. The multiplicity M IMF will growas a function of temperature, will reach a peak and then begin to go down as, at a hightemperature, only light particles are dominant. For evidence and discussion of this see [6].For projectile fragmentation we are in the domain where M IMF rises with temperature. Nowat constant temperature, let us consider what must happen if the dissociating system growsbigger. We expect M IMF will increase with the size of the dissociating system, that is, with Z bound . Experimental data are quite different: M IMF initially increases, reaches a maximumat a particular value of Z bound and then goes down.In Fig.1 we show two graphs for M IMF , one in which the temperature is kept fixed (at6.73 MeV) and another in which T decreases linearly from 7.5 MeV (at b =0)to 3 MeV at b max . The calculation is qualitative. The case considered is Sn on
Sn. CTM is usedto calculate M IMF but evaporation is not included. Simlarly Z bound is Z s (no correction for z = 1 particles). Fuller calculations will be shown later but the principal effects are all inthe graphs. Keeping the temparature fixed makes M IMF go up all the way till Z bound = Z isreached. One needs the temperature to go down to bring down the value of M IMF as seenin experiment.
IV. USE A MODEL TO EXTRACT b -DEPENDENCE OF TEMPERATURE In our model we can use an iterative technique to deduce a temperature from experi-mental data of M IMF vs Z bound . Pick a b ; abrasion gives a < Z s > . Guess a temperature T . A full calculation with CTM and evaporation is now done to get a Z bound and M IMF .This Z bound will be close to < Z s > . If the guessed value of temperature is too low then thecalculated value of M IMF will be too little for this value of Z bound when confronted withdata. In the next iteration the temperature will be raised. If on the other hand, for theguess value of T , the calculated M IMF is too high, in the next iteration the temperaturewill be lowered. Of course when we change T , calculated Z bound will also shift but thischange is smaller and with a small number of iterations one can approximately reproducean experimental pair Z bound , M IMF .For the case of
Sn on
Sn we provide Table I which demonstrates this. The firsttwo columns are data from experiment. The next two columns are the values of Z bound and4 xperimental Theoretical Z bound M IMF Z bound M IMF b Required T(fm) (MeV)11.0 1.421 11.080 1.424 2.912 6.39815.0 1.825 15.094 1.818 3.625 6.10820.0 2.145 19.984 2.131 4.4574 5.84025.0 2.010 25.024 2.019 5.289 5.52030.0 1.505 29.854 1.545 6.122 5.25035.0 0.920 34.985 0.928 7.072 4.97040.0 0.415 39.639 0.424 8.023 4.65045.0 0.193 44.763 0.196 9.331 4.35047.0 0.156 46.512 0.154 9.925 4.26049.0 0.135 48.425 0.130 10.876 4.190TABLE I: Best fit and experimental values for
Sn on
Sn. The first two columns are data fromexperiment. The next two columns are the values of Z bound and M IMF we get from our iterativeprocedure. These values are taken to be close enough to the experimental pair. These are obtainedfor a value of b (fifth column) and a temperature T (sixth column). M IMF we get from our iterative procedure. These values are taken to be close enough to theexperimental pair. These are obtained for a value of b (sixth column) and a temperature T (fifth column). Table II provides similar compilation for Sn on
Sn.Having deduced once for all such “experimental data” of T vs b , one can try simple parametri-sation like T ( b ) = C + C ∗ b + C ∗ b ... . and see how well they fit the data. We show thisfor the two cases in Fig.2.In Fig.3 using such parametrised versions of T we compute M IMF vs Z bound and comparewith experimental data. Except for fluctuations in the values of M IMF for very low valuesof Z bound the fits are very good. We will return to the cases of fluctuations in a later section.5 xperimental Theoretical Z bound M IMF Z bound M IMF b Required T(fm) (MeV)15.0 1.690 14.816 1.583 3.886 6.20020.0 1.923 19.865 1.906 4.698 5.74021.0 1.984 21.207 1.976 4.930 5.70525.0 1.749 24.913 1.758 5.510 5.32030.0 1.079 30.356 1.075 6.438 4.90035.0 0.581 35.252 0.602 7.366 4.60040.0 0.223 40.123 0.225 8.410 4.21045.0 0.201 44.676 0.199 9.802 4.10047.0 0.201 47024 0.159 10.876 4.000TABLE II: Same as Table 1, except that here the projectile is
Sn instead of
Sn.
V. TEMPERATURES EXTRACTED FROM ISOTOPE POPULATIONS
In the preceding sections we have extracted temperatures T (combining data and model)at values of b (equivalently at values of Z bound ). This is a new method for extracting temper-ature. A more standard way of extracting temperatures is the Albergo formula [9] which hasbeen widely used in the past (for a review see, for example, [10, 11]). In [[7], Figs.24 and 25]temperatures at selected values of Z bound /Z were extracted from populations in [ , He, , Li]and [ , Be, , Li] using Albergo formula. These temperatures are compared in Fig.4 with atypical temperature profile deduced here. It is gratifying to see that such different methodsof extraction still give reasonable agreement.
VI. FLUCTUATIONS IN M IMF
FOR SMALL Z bound For small values of Z bound the measured M IMF shows considerable fluctuations as we gofrom one value of Z bound to another (see Fig.3). Our model does not reproduce these althoughgeneral shapes are correct. Statistical models are not expected to show such fluctuationsbut let us get into some details which (a) give a clue how such fluctuations may arise and6b) why our model misses them. The reader who is not interested in such details can skipthe rest of this section without loss of continuity.For definiteness, consider the case of Sn on
Sn. By the definition of
IM F, ( z > M IMF is 0 when Z bound is 2. Consider now Z bound = 3. The most direct way one can havethis is if the PLF has Z s =3. Taking very simplistic point of view, suppose, this also has N s = 3, that is, the PLF is Li. This is stable and we immediately get M IMF = 1. This isindeed the experimental value. The case of Z bound =4 may arise if the PLF is Z s = 4 , N s = 4,i.e., if the PLF is Be. But Be is unbound and will break up into two α particles which arenot IM F ’s. Thus M IMF drops to zero. In experiment this falls to about 0.3 rather than 0.The simple fact that Li (and an excited state of Li) is particle stable whereas the statesof Be are not is not embedded in our liquid-drop model for ground state and Fermi-gasmodel for excited states. Our description gets better for larger nuclear systems but for verysmall systems quantum mechanics of nuclear forces causes rapid changes in properties asone goes from one excited state to another and one nucleus to another. Our model can notaccommodate this.Let us go back to the case of Z s =3 and treat it more realistically. Using the abraisonmodel, when Z s is 3, PLF can have N s =3,4 and 5. Probabilities for higher and lower valuesof N s are small. Following our model we get M IMF ≈ .
94 with Z bound slightly less than 3.When Z s is 4, significant probabilities occur for N s =5,6 and 7. Following our model we geta small increase in M IMF with Z bound whereas experimentally M IMF falls. This discrepancyhappens because in the Fermi-gas model there is very little difference between propertiesof ground and excited states of Li and Be whereas, in reality they are very different. Amuch more ambitious calculation for very small dissociating systems with Z s between 3 and7 where we take binding energies and values of excited state energies from experiments (thisbecomes more and more unwieldy as Z s increases) is under way.Fig.7 in ref [7] shows that SMM calculations are able to reproduce the fluctuations faith-fully. Actually in those calculations the occurrences of Z s , N s with associated E x are notcalculated but guessed so that the ensemble produces the data as faithfully as possible. Forfurther details how these calculations were done please refer to [8].7 II. TOWARDS A UNIVERSAL TEMPERATURE PROFILE
Knowing the temperature profile T = T ( b ) in one case, say Sn on
Sn, can weanticipate what T = T ( b ) will be like in another case, say, Ni on Be ? In both the cases b min is zero and b max is R + R yet we can not expect the same functional form T = T ( b/b max )for both the cases. In the first case, near b = 0 a small change in b causes a large fractionalchange in the mass of the PLF whereas, for Ni on Be, near b =0, a small change in b causes very little change in the mass of the PLF. Thus we might expect the temperature tochange more rapidly in the first case near b =0, whereas, in the second case, the temperaturemay change very little since not much changed when b changed a little. In fact, for Ni onBe, transport model calculations, HIPSE (Heavy Ion Phase Space Exploration) and AMD(Antisymmetrised Molecular Dynamics) find that starting from b =0, excitation energy/perparticle changes very little in the beginning [13]. In terms of our model, this would meanthat for Ni on Be, T would be slow to change in the beginning.We might argue that a measure of the wound that the projectile suffers in a heavy ioncollision is 1 . − A s /A and that the temperature depends upon the wound. Thus we shouldexpect T = T ( A s ( b ) /A ). Just as we can write T ( b ) = C + C ∗ b + C ∗ b + ... so also wecould expand in powers of A s ( b ) /A ,i.e., T ( b ) = D + D ( A s ( b ) /A ) + D ( A s ( b ) /A ) + ... We try such fits to the “experimental” temperature profile given in Tables I and II. From b we deduce A s ( b ) /A and plot T as a function of A s ( b ) /A . A linear fit appears to be goodenough (Fig.5).The specification that T ( b ) = D + D ( A s ( b ) /A ) has profound consequences. This meansthe temperature profile T ( b/b max ) of Sn on
Sn is very different from that of Ni on Be. In the first case A s ( b ) /A is nearly zero for b = b min =0 whereas in the latter case A s ( b ) /A is ≈ . b = b min =0. For D =7.5 MeV and D =-4.5 MeV, the temperatureprofiles are compared in Fig.6. Even more remarkable feature is that the temperature profileof Ni on Be is so different from the temperature profile of Ni on
Ta. In the lattercase b min = R T a − R Ni and beyond b min , A s ( b ) /A grows from zero to 1 for b max . This isvery similar to the temperature profile of Sn on
Sn.8
III. FORMULAE FOR CROSS-SECTIONS
Now that we have established that temperature T should be considered impact parameter b dependent, let us write down how cross-sections should be evaluated. We first start withabrasion cross-section. In eq.(1) of [1], the abrasion cross-section was written as σ a,N s ,Z s = 2 π Z bdbP N s ,Z s ( b ) (1)where P N s ,Z s ( b ) is the probability that a PLF with N s neutrons and Z s protons emergesin collision at impact parameter b . Actually there is an extra parameter that needs to bespecified. The complete labelling is σ a,N s ,Z s ,T if we assume that irrespective of the value of b , the PLF has a temperature T . Here we have broadened this to the more general casewhere the temperature is dependent on the impact parameter b . Thus the PLF with N s neutrons and Z s protons will be formed in a small range of temperature (as the productionof a particular N s , Z s occurs in a small range of b ).To proceed, let us discretize. We divide the interval b min to b max into small segments oflength ∆ b . Let the mid-point of the i -th bin be < b i > and the temperature for collision at < b i > be T i . Then σ a,N s ,Z s = X i σ a,N s ,Z s ,T i (2)where σ a,N s ,Z s ,T i = 2 π < b i > ∆ bP N S ,Z s ( < b i > ) (3)PLF’s with the same N s , Z s but different T i ’s are treated independently. The rest of thecalculation proceeds as in [1]. If, after abrasion, we have, a system N s , Z s at temperature T i ,CTM allows us to compute the average population of the composite with neutron number n ,proton number z when this system breaks up (this composite is at temperature T i ). Denotethis by M N s ,Z s ,T i n,z . It then follows, summing over all the abraded N s , Z s that can yield n, z the primary 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 (4)Finally, evaporation from these composites n, z at temperatures T i is considered beforecomparing with experimental data. 9 X. CROSS-SECTIONS FOR DIFFERENT REACTIONS
We will now show some results for cross-sections using our model and compare withexperimental data. We first show results for
Sn on
Sn and
Sn on
Sn at 600MeV/nucleon beam energy. The experimental data are plotted in [7] and the data weregiven to us, thanks to Prof. Trautmann. The differential charge distributions and isotopicdistributions for
Sn on
Sn and
Sn on
Sn were theoretically calculated using T ( b ) = C + C b and and also T ( b ) = D + D ( A s ( b ) /A ). So long as the temperature values atthe two end points of b are the same, the answers did not differ much. In Fig.7 we haveshown results for T varying linearly with b with T max = 7 . T min =3 MeV. At each Z bound , the charge distribution and isotopic distributions are calculated separately and finallyintegrated over different Z bound ranges. The differential charge distributions are shown inFig.7 for different intervals of Z bound /Z ranging between 0 . .
2, 0 . .
4, 0 . . . . . .
0. For the sake of clarity the distributions are normalized withdifferent multiplicative factors. At peripheral collisions (i.e. 0 . ≤ Z bound /Z ≤ .
0) due tosmall temperature of the projectile spectator, it breaks into one large fragment and smallnumber of light fragments, hence the charge distribution shows U type nature. But with thedecrease of impact parameter the temperature increases, the projectile spectator breaks intolarge number of fragments and the charge distributions become steeper. In Figs.8 and 9 theintegrated isotopic distributions over the range 0 . ≤ Z bound /Z ≤ . Sn on
Snand
Sn on
Sn reaction respectively.Rest of the cross-sections shown all use T ( b ) = 7 . A S ( b ) /A )4 . Ni on Be and
Ta at beam energy 140 MeV/nucleon done at MichiganState University. The data were made available to us by Dr.Mocko (Mocko, Ph.D. thesis).Calculations were also done with Ni as beam. Those results agree with experiment equallywell but are not shown here for brevity. The results for Ni on Be and Ni on
Ta areshown in Figs.11 to 14. The experimental data are from [13]. The chief difference fromresults shown in [1] is that we are able to include data for very peripheral collisions. Nextwe look at some older data from
Xe on Al at 790 MeV/nucleon [14]. Results are givenin Figs. 15 and 16. 10he parametrization T ( b ) = 7 . A s ( b ) /A )4 . M IMF vs. Z bound for Sn isotopes is foundwith slightly different values: T ( b ) = 7 . A s ( b ) /A )3 . X. SUMMARY AND DISCUSSION
We have shown that there are specific experimental data in projectile fragmentation whichclearly establish the need to introduce an impact parameter dependence of temperature T in the PLF formed. Combining data and a model one can establish approximate values of T = T ( b ). The model for cross-sections has been extended to incorporate this temperaturevariation. This has allowed us to investigate more peripheral collisions. In addition, theimpact parameter dependence of temperature appears to be very simple: T ( b ) = D + D ( A s ( b ) /A ) where D and D are constants, A s ( b ) is the mass of the PLF and A is themass of the projectile. With this model, we plan to embark upon an exhaustive study ofavailable data on projectile fragmentation. XI. ACKNOWLEDGMENTS
This work was supported in part by Natural Sciences and Engineering Research Councilof Canada. The authors are thankful to Prof. Wolfgang Trautmann and Dr. M. Mockofor access to experimental data. S. Mallik is thankful for a very productive and enjoyablestay at McGill University for part of this work. S. Das Gupta thanks Dr. Santanu Pal forhospitality at Variable Energy Cyclotron Centre at Kolkata. [1] S. Mallik, G. Chaudhuri and S. Das Gupta, Phys. Rev.
C 83 , 044612 (2011).[2] S. Das Gupta and A. Z. Mekjian,Phys. Rep. , 131 (1981).[3] C. B. Das, S. Das Gupta, W. G. Lynch, A. Z. Mekjian and M. B. Tsang, Phys. Rep. , 1(2005).[4] G. Chaudhuri, S. Mallik, Nucl. Phys. A 849 , 190 (2011).[5] C. Sfienti et al., Phys. Rev. Lett. , 152701 (2009).[6] M. B. Tsang et al., Phys. Rev. Lett. , 1502 (1993).
10 20 30 40 500369 Sn +Sn Z s < M I M F > p r i T=7.5 to 3.0 MeV
T=6.73 MeV
FIG. 1: (Color Online) Mean multiplicity of intermediate-mass fragments M IMF (after multifrag-mentation stage), as a function of projectile spectator charge for
Sn on
Sn reaction calculatedat a fixed temperature T =6.73 MeV (black solid line) and at a linearly decreasing temperaturefrom 7.5 MeV at b =0 to 3 MeV at b max (red dotted line). The ordinate is labelled < M IMF > pri as the effect of evaporation is not included.[7] R. Ogul et al.,Phys. Rev C 83 ,024608(2011)[8] A. S. Botvina et al., Nucl. Phys.
A 584 , 737 (1995).[9] S. Albergo et al., Il Nuovo Cimento , A1(1985)[10] S. Das Gupta, A. Z. Mekjian and M. B. Tsang, Advances in Nuclear Physics,Vol.26,89(2001)edited by J. W. Negele and E. Vogt, Plenum Publishers, New York.[11] J. Pochodzalla and W. Trautmann, Isospin Physics in Heavy-Ion Collisions at Intermedi-ate Energies, 451(2001) edited by B-A Li and W. U. Schr¨ o der, Nova Science Publishers,Inc,Huntington, New York.[12] M. Mocko,Ph.D. thesis, Michigan State University, 2006.[13] M. Mocko et al., Phys. Rev. C 78 , 024612 (2008).[14] J. Reinhold et al., Phys. Rev.
C 58 , 247 (1998). T=C +C *b C =7.50 C =-0.3786 Sn +Sn T=C +C *b C =7.50 C =-0.3947 Sn +Sn T=C +C *b C =7.10 C =-0.2879 T e m p e r a t u r e ( M e V ) T=C +C *b C =7.07 C =-0.3024 T=C +C *b+C *b C =8.19 C =-0.6623 C =0.0025 T=C +C *b+C *b C =7.83 C =-0.5342 C =0.0175 Impact Parameter (fm)
FIG. 2: (Color Online) Impact parameter dependence of temperature for
Sn on
Sn (leftpanels) and
Sn on
Sn reactions (right panels). The red squires in the upper panels representthe extracted temperatures (sixth column of table-I and II) and the blue dotted lines are linearlydecreasing temperature profile from 7.5 MeV to 3 MeV. The blue dotted lines of middle andlower panels represent fitting of extracted temperatures (red squires) with T ( b ) = C + C ∗ b and T ( b ) = C + C ∗ b + C ∗ b equation respectively. The unit of C is MeV, C is MeVfm − and C is MeVfm − .
10 20 30 40 < M I M F > Sn +Sn Sn +Sn Z bound FIG. 3: (Color Online) Mean multiplicity of intermediate-mass fragments M IMF , as a functionof Z bound for Sn on
Sn (left panel) and
Sn on
Sn (right panel) reaction calculated usinglinearly decreasing temperature from 7.5 MeV to 3 MeV (red solid lines) and T ( b ) = C + C ∗ b + C ∗ b profile (blue dotted lines). The experimental results are shown by the black dashed lines.
10 20 30 40024 Sn +Sn T e m p e r a t u r e ( M e V ) Z bound Sn +Sn FIG. 4: (Color Online) Comparison of theoretically used temperature profiles (i) temperaturedecreasing linearly with impact parameter from 7.5 MeV to 3 MeV (red solid lines), (ii) T ( b ) = C + C ∗ b + C ∗ b fitting temperature (blue dotted lines) with that deduced by Albergo formulafrom experimental data (black points with error bars) for Sn on
Sn (left panel) and
Sn on
Sn (right panel). T=D +D *(A s /A ) D =7.20 D =-3.02 T=D +D *(A s /A ) D =7.24 D =-3.39 Sn +Sn T e m p e r a t u r e ( M e V ) Sn +Sn T=D +D *(A s /A )+D *(A s /A ) D =7.17 D =-3.18 D =0.132 T=D +D *(A s /A )+D *(A s /A ) D =8.22 D =-6.61 D =2.366 A s /A FIG. 5: (Color Online) Fitting of extracted temperatures (red squires) with T ( b ) = D + D ( A s ( b ) /A ) (blue dotted lines in upper panels) and T ( b ) = D + D ( A s ( b ) /A ) + D ( A s ( b ) /A ) profile (blue dotted lines in lower panels) for Sn on
Sn (left panels) and
Sn on
Sn (rightpanels). The units of D , D and D are MeV. .0 0.2 0.4 0.6 0.8 1.0345678 Ni +Be Ni +Ta Sn +Sn T e m p e r a t u r e ( M e V ) (b-b min )/(b max -b min ) FIG. 6: (Color Online) Temperature profile for Ni on Be (black solid line), Ni on
Ta (reddotted line) and
Sn on
Sn (blue dashed line) by considering T = 7 . − . A s ( b ) /A )
10 20 30 4010 -6 -3 -6 -3 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 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 =0.8-1.0x10 x10 x10 x10 -2 x10 -4 Sn +Sn FIG. 7: (Color Online) Theoretical total charge cross-section distribution (red solid lines) for
Sn on
Sn (left panel) and
Sn on
Sn reaction (right panel) sorted into five intervals of Z bound /Z ranging between 0 . .
2, 0 . .
4, 0 . .
6, 0 . . . . − , 10 − , 10 , 10 , 10 respectively. The experimental data are shown byblack dashed lines. Theoretical calculation is done using linearly decreasing temperature from 7.5MeV at b =0 to 3 MeV at b max . -3 -1 Z=4 -3 -1 Z=6 -3 0 3 610 -3 -1 Z=8Neutron Excess (N-Z) -3 0 3 6 10 -3 -1 Z=10 C r o ss - sec t i on ( m b ) FIG. 8: (Color Online) Theoretical isotopic cross-section distribution (circles joined by dashedlines) for
Sn on
Sn reaction summed over 0 . ≤ Z bound /Z ≤ .
8. The experimental data areshown by black squires. Theoretical calculation is done using linearly decreasing temperature from7.5 MeV at b =0 to 3 MeV at b max . -3 -1 Z=4 -3 -1 Z=6 -3 0 3 610 -3 -1 Z=8 -3 0 3 6 -3 -1 C r o ss - sec t i on ( m b ) Z=10
Neutron Excess (N-Z)
FIG. 9: (Color Online) Same as Fig. 8, except that here the projectile is
Sn instead of
Sn.
10 20 30 4010 -6 -3 -6 -3 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 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 Sn +Sn FIG. 10: (Color Online) Same as Fig. 7 except that here the temperature profile is T ( b ) =7 . M eV − ( A S ( b ) /A )4 . M eV instead of linearly decreasing temperature from 7.5 MeV at b =0 to3 MeV at b max
10 15 20 25 3010 -2
10 20 30 40 5010 -2 C r o ss - sec t i on ( m b ) Mass Number (A) Proton Number (Z)
FIG. 11: (Color Online) Total mass (left panel) and total charge (right panel) cross-sectiondistribution for the Ni on Be reaction. The left panel shows the cross-sections as a functionof the mass number, while the right panel displays the cross-sections as a function of the protonnumber. The theoretical calculation is done using temperature decreasing linearly with A s /A from 7.5 MeV to 3.0 MeV (dashed line) and compared with the experimental data (solid line).
20 30 40 5010 -2 -2 Proton Number (Z)Mass Number (A) C r o ss - sec t i on ( m b ) FIG. 12: (Color Online) (Color Online) Same as Fig. 11 except that here the target is
Tainstead of Be. -7 -5 -3 -1 Z=6 Z=9
Z=12 -7 -5 -3 -1 Z=15 -7 -5 -3 -1 Z=18 C r o ss - sec t i on ( m b ) Z=21
Z=24
Neutron Excess (N-Z) -7 -5 -3 -1 Z=27
FIG. 13: (Color Online) Theoretical isotopic cross-section distribution (circles joined by dashedlines) for Ni on Be reaction compared with experimental data (squares with error bars). -7 -5 -3 -1 Z=6
Z=9
Z=12 -7 -5 -3 -1 Z=15 -7 -5 -3 -1 Z=18
Z=21 C r o ss - sec t i on ( m b ) Neutron Excess (N-Z)
Z=24 -7 -5 -3 -1 Z=27
FIG. 14: (Color Online) Same as Fig. 13 except that here the target is
Ta instead of Be. -2 Proton Number (Z) C r o ss - sec t i on ( m b ) FIG. 15: (Color Online) Total charge cross-section distribution for the
Xe on Al reaction. Thetheoretical calculation is done using temperature decreasing linearly with A s /A from 7.5 MeV to3.0 MeV (dashed line) and compared with the experimental data (solid line). -7 -5 -3 -1 Z=40 -7 -5 -3 -1 Z=43 -7 -5 -3 -1 Z=46 -7 -5 -3 -1 C r o ss - sec t i on ( m b ) Neutron Excess (N-Z)Z=49
FIG. 16: (Color Online) Theoretical isotopic cross-section distribution (circles joined by dashedlines) for