Effect of secondary decay on isoscaling: Results from the canonical thermodynamical model
aa r X i v : . [ nu c l - t h ] J u l Effect of secondary decay on isoscaling: Results from thecanonical thermodynamical model
Gargi Chaudhuri and Swagata Mallik
Variable Energy Cyclotron Centre, 1/AF Bidhannagar, Kolkata 700064 (Dated: October 29, 2018)
Abstract
The projectile fragmentation reactions using N i & N i beams at 140 MeV/n on targets Be & T a are studied using the canonical thermodynamical model coupled with an evaporationcode. The isoscaling property of the fragments produced is studied using both the primary and thesecondary fragments and it is observed that the secondary fragments also respect isoscaling thoughthe isoscaling parameters α and β changes. The temperature needed to reproduce experimentaldata with the secondary fragments is less than that needed with the primary ones. The canonicalmodel coupled with the evaporation code successfully explains the experimental data for isoscalingfor the projectile fragmentation reactions. PACS numbers: 25.70Mn, 25.70Pq . INTRODUCTION Projectile fragmentation reaction is used extensively to study the reaction mechanisms inheavy ion collisions at intermediate and high energies. This is also an important techniquefor the production of rare isotope beams and is used by many radioactive ion beam facilitiesaround the world. The fragment cross sections of projectile fragmentation reactions usingprimary beams of Ca , Ca , N i and N i at 140 MeV/nucleon on Be and T a targetshave been measured at the National Superconducting Cyclotron Laboratory at MichiganState University[1]. The canonical thermodynamical model(CTM)[2] has been used to cal-culate some of these fragment cross sections[3]. In the present work, an evaporation code hasbeen developed and has been coupled with the canonical thermodynamical model. CTMcoupled with this secondary decay code is then used to analyze the isoscaling data fromthe projectile fragmentation reactions. The lighter fragments produced from these reactionsexhibit the linear isoscaling[4–6] phenomena and our model calculation also strongly sup-ports this observation. The secondary fragments (produced after applying the evaporationcode on the canonical model) also exhibit isoscaling like the primary fragments from thefragmentation reaction[7] but the temperature needed to reproduce the experimental datawith the secondary fragments is lower than that required by the calculation with the pri-mary fragments. The isoscaling parameters α and β as obtained in the present work fromthe model calculations agree closely with those obtained from the experimental data. Theseparameters as obtained from the secondary fragments are lower in magnitude than thoseobtained from the primary ones. This effect is also seen in the dynamical models [8] thoughthe reduction is much more there. The isoscaling behaviour displayed by the fragmentsproduced in the projectile fragmentation reactions and the effect of evaporation on it in theframework of the HIPSE model has been discussed recently in [9].This paper is structured as follows. First we describe the canonical model briefly inSec.II. In the same section we also present the main features of the evaporation code whichis used to calculate the secondary fragments. In Sec.III we present the results. The effectof sequential decay on the distribution of the isotopic fragments is discussed. The isoscalingphenomena as displayed by the primary as well as the secondary fragments is also describedin this section and are compared with the experimental data. In Sec. IV we present thesummary. 2 I. THE STATISTICAL MODEL
In models of statistical disassembly of a nuclear system formed by the collision of twoheavy ions at intermediate energy one assumes that because of multiple nucleon-nucleoncollisions a statistical equilibrium is reached. Consequently, the temperature rises. Thesystem expands from normal density and composites are formed on the way to disassembly.As the system reaches between three to six times the normal volume, the interactions betweencomposites become unimportant (except for the long range Coulomb interaction) and onecan do a statistical equilibrium calculation to obtain the yields of composites at a volumecalled the freeze-out volume. The partitioning into available channels can be solved in thecanonical ensemble where the number of particles in the nuclear system is finite (as it wouldbe in experiments). In the next subsection we describe the canonical model.
A. The canonical thermodynamical model
In this section we describe briefly the canonical thermodynamical model. Assume that thesystem with A nucleons and Z protons at temperature T , has expanded to a higher thannormal volume and the partitioning into different composites can be calculated accordingto the rules of equilibrium statistical mechanics. In a canonical model, the partitioning isdone such that all partitions have the correct A , Z (equivalently N , Z ). Details of theimplementation of the canonical model can be found elsewhere [2]; here we give the essentialsnecessary to follow the present work.The 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 channels isenormous) which satisfy N = P I × n I,J and Z = P J × n I,J ; ω I,J is the partition functionof one composite with neutron number I and proton number J respectively and n I,J is thenumber of this composite in the given channel. The one-body partition function ω I,J is aproduct of two parts: one arising from the translational motion of the composite and anotherfrom the intrinsic partition function of the composite: ω I,J = V f h (2 πmT ) / A / × z I,J ( int ) (2)3ere A = I + J is the mass number of the composite and V f is the volume available fortranslational motion; V f will be less than V , the volume to which the system has expandedat break up. We use V f = V − V , where V is the normal volume of nucleus with Z protonsand N neutrons. In this calculation we have used a fairly typical value V = 6 V .The probability of a given channel P ( ~n I,J ) ≡ P ( n , , n , , n , ......n I,J ....... ) is given by P ( ~n I,J ) = 1 Q N ,Z Y ω n I,J
I,J n I,J ! (3)The average number of composites with I neutrons and J protons is seen easily from theabove equation to be h n I,J i = ω I,J Q N − I,Z − J Q N ,Z (4)The constraints N = P I × n I,J and Z = P J × n I,J can be used to obtain different lookingbut equivalent recursion relations for partition functions[10]. For example Q N ,Z = 1 N X I,J Iω I,J Q N − I,Z − J (5)These recursion relations allow one to calculate Q N ,Z We list now the properties of the composites used in this work. The proton and theneutron are fundamental building blocks thus z , ( int ) = z , ( int ) = 2 where 2 takes careof 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 of the compositeand (2 s I,J + 1) is the experimental spin degeneracy of the ground state. Excited states forthese very low mass nuclei are not included. For mass number A = 5 and greater we usethe 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 ] (6)The derivation of this equation is given in several places [2, 11] so we will not repeat thearguments here. The expression includes the volume energy, the temperature dependentsurface energy, the Coulomb energy and the symmetry energy. The term T Aǫ representscontribution from 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.(17)). For I, J ,(the neutron and the proton number) we include a ridge along the line of stability. The4iquid-drop formula above also gives neutron and proton drip lines and the results shownhere include all nuclei within the boundaries.The long range Coulomb interaction between different composites can be included in anapproximation called the Wigner-Seitz approximation. We incorporate this following thescheme set up in [11].
B. The evaporation code
The statistical multifragmentation model described above calculates the properties of thecollision averaged system that can be approximated by an equilibrium ensemble. Ideally,one would like to measure the properties of excited primary fragments after emission inorder to extract information about the collisions and compare directly with the equilibriumpredictions of the model. However, the time scale of a nuclear reaction(10 − s ) is muchshorter than the time scale for particle detection (10 − s ). Before reaching the detectors,most fragments decay to stable isotopes in their ground states. Thus before any modelsimulations can be compared to experimental data, it is indispensable to have a modelthat simulates sequential decays. A Monte Carlo technique is employed to follow all decaychains until the resulting products are unable to undergo further decay. For the purposes ofthe sequential decay calculations the excited primary fragments generated by the statisticalmodel calculations are taken as the compound nucleus input to the evaporation code. Hence,every primary fragment is decayed as a separate event.We consider the deexcitation of a primary fragment of mass A , charge Z and temperature T . The succseesive particle emission from the hot primary fragments is assumed to be thebasic deexcitation mechanism. For each event of the primary breakup simulation, the entirechain of evaporation and secondary breakup events is Monte Carlo simulated. The standardWeisskopf evaporation scheme is used to take into account evaporation of nucleons, d , t , He and α . The decays of particle stable excited states via gamma rays were also taken intoaccount for the sequential decay process and for the calculation of the final ground stateyields. We have also considered fission as a deexcitation channel though for the nuclei ofmass <
100 its role will be quite insignificant. The process of light particle emission froma compound nucleus is governed by the emission width Γ ν at which a particle of type ν isemitted. According to Weisskopf’s conventional evaporation theory [12], the partial decay5idth for emission of a light particle of type ν is given byΓ ν = gmσ π ¯ h ( E ∗ − E − V ν ) a R exp(2 q a R ( E ∗ − E − V ν ) − q a P E ∗ ) (7)Here m is the mass of the emitted particle, g is its spin degeneracy. E is the particle sep-aration energy which is calculated from the binding energies of the parent nucleus, daughternucleus and the binding energy of the emitted particle and the liquid drop model is used tocalculate the binding energies. The subscript ν refers to the emitted particle, P refers to theparent nuclei and R refers to the residual(daughter) nuclei. a P & a R are the level densityparameters of the parent and residual nucleus respectively. The level density parameter isgiven by a = A/ M eV − and it connects the excitation energy E ∗ and temperature T through the following relations. E ∗ = a P T P ( E ∗ − E − V ν ) = a R T R . (8)where T P & T R are the temperatures of the emitting(parent) and the final(residual)nucleus respectively. V ν is the Coulomb barrier which is zero for neutral particles and non-zero for charged particles. In order to calculate the Coulomb barrier for charged particles ofmass A ≥ V ν = Z ν ( Z P − Z ν ) e r i { A / ν + ( A P − A ν ) / } for A ν ≥
2= ( Z P − e r i A / P for protons (9)where r i is taken as 1.44m. σ is the geometrical crosssection (inverse cross section) associated with the formation ofthe compound nucleus(parent) from the emitted particle and the daughter nucleus and isgiven by σ = πR where, R = r { ( A P − A ν ) / + A ν / } for A ν ≥ r ( A P − / for A ν = 1 . (10)6here r = 1.2 fm.For the emission of giant dipole γ -quanta we take the formula given by Lynn[14]Γ γ = 3 ρ P ( E ∗ ) Z E ∗ dερ R ( E ∗ − ε ) f ( ε ) (11)with f ( ε ) = 43 π κm n c e ¯ hc N P Z P A P Γ G ε (Γ G ε ) + ( ε − E G ) (12)with κ = 0 .
75, and E G and Γ G are the position and width of the giant dipole resonance.For the fission width we have used the simplified formula of Bohr-Wheeler given byΓ f = T P π exp ( − B f /T P ) (13)where B f is the fission barrier of the compound nucleus given by[15] B f ( M eV ) = − . Z P + 0 . A P − Z P ) + 101 . . (14)Once the emission widths are known, it is required to establish the emission algorithmwhich decides whether a particle is being emitted from the compound nucleus. This isdone [16] by first calculating the ratio x = τ /τ tot where τ tot = ¯ h/ Γ tot , Γ tot = P ν Γ ν and ν = n, p, d, t, He , α, γ or fission and then performing Monte-Carlo sampling from a uniformlydistributed set of random numbers. In the case that a particle is emitted, the type of theemitted particle is next decided by a Monte Carlo selection with the weights Γ ν / Γ tot (partialwidths). The energy of the emitted particle is then obtained by another Monte Carlosampling of its energy spectrum. The energy, mass and charge of the nucleus is adjustedafter each emission. This procedure is followed for each of the primary fragment producedat a fixed temperature and then repeated over a large ensemble and the observables arecalculated from the ensemble averages. . The number and type of particles emitted and thefinal decay product in each event is registered and are taken into account properly keepingin mind the overall charge and baryon number conservation. III. RESULTS
First we will show our calculations for N i on Be reaction and N i on Be reaction. Inthe model, the target imparts a certain amount of energy to the projectile transforming it to7 projectile like fragment(PLF) with a temperature. This excited PLF will then expand andform composites during the expansion. The partioning of the PLF into different compositesis done by the rules of equilibrium statistical mechanics in a freeze-out volume. We considerproduction of different isotopes from the statistical breakup of the dissociating system. If < n i,j > is the average number(multiplicity) of composites with i neutrons and j protons,then the cross-section for this composite is σ ( i, j ) = C < n i,j > , where C is a constant notcalculable from the thermodynamic model. It depends upon the dynamics that are outsidethe scope of this model. To be able to compute < n i,j > we need to know the mass andcharge of the PLF and its temperature. The source sizes adopted for this calculation are zeroorder guesses. It could be sometimes smaller or greater depending on the diffusion from thetarget. For N i or N i on Be which is a small target the choice of the mass and the chargeof the PLF is limited. It can be slightly less than that of the projectile to as large as thatof the projectile plus Be , the last being the case when the much larger projectile swallowsthe small target and drags it along retaining PLF features. Similarly we have some limitson energy imparted(this fixes the temperature). This energy can be small or upto the upperlimit. The upper limit is given by the case of projectile swallowing Be and all the energytransforming into internal excitation(no part going into collective flow). In the canonicalcalculation, the dissociating system is taken to be N i + Be ( N = 35 , Z = 32) and forthe other reaction the dissociating system is taken to be N i + Be ( N = 41 , Z = 32).Allcomposites between drip lines are included as detailed in Sec.IIA with the highest values ofN, Z terminating at N , Z . The temperature is taken to be 5.8 MeV for both the reactions.Fig. 1 displays the isotopic distribution for Z=12(magnesium) produced from both thethe reactions. The dashed lines correspond to the distributions of the primary fragmentswhile the solid lines correspond to the distributions after sequential decay. As expected,the more neutron rich system with N /Z = 1 .
28 (right panel) produces more neutron richisotopes than the neutron deficient system with N /Z = 1 .
09 (left panel). In all cases, theprimary distributions are much wider and more neutron rich than the final distributions.The peak positions of the isotopic distributions of both the primary and the secondaryfragments coincide in case of the neutron deficient system as seen from the left panel ofthe figure. In case of the neutron rich system(right panel) the peak of the distribution ofthe secondary fragments has shifted to the left with respect to that of the primary. Theexperimental isotopic distributions(solid squares with error bars) agree much more with the8nal results obtained after secondary decay than with the primary distributions. The widthand peak position of the isotopic distribution after the secondary decay agrees very well withthe experimental data. The model also successfully reproduces the rapid fall in crosssectionfor large neutron number.We will now discuss the results about isoscaling. It is observed from the experimentaldata[17] that the light fragments emitted from the N i and N i systems exhibit the linearisoscaling behaviour represented by the equation R = Y ( N, Z ) /Y ( N, Z ) = C exp( αN + βZ ) . (15)where the isoscaling ratio R ( N, Z ) is factored into two fugacity terms α and β , whichcontain the differences of the chemical potentials for neutrons and protons of the two reactionsystems. Y ( N, Z ) refers to the yield of fragment(N,Z) from system 2 which is usuallytaken to be the neutron-rich one and Y ( N, Z ) refers to the same from system 1. C is anormalization factor of the isoscaling ratio. It is observed from our model that both theprimary as well as the secondary fragments exhibit isoscaling. Fig. 2 shows the isoscalingresults for Ni on Be system for the primary fragments. The ratio R is plotted as functionof the neutron number from Z =6 to Z= 13 in the left panel whereas the right panel displaysthe ratio as function of the proton number Z from N=8 to N=15. It is seen that the primaryfragments exhibit very well the linear isoscaling behaviour for the lighter fragments over awide range of isotopes and isotones. The lines in the figures are the best fits of the calculated R ratios(open triangles) to Eq.15. They are essentially linear and parallel on the semi logplot .Fig. 3 displays the isoscaling results for the secondary fragments. The open trianglesare the results obatined from our model while the solid squares with the error bars are theexperimental ratios. We have shown the isoscaling results for the even Z and odd Z isotopesin two separate panels for the sake of clarity. While comparing with the results of the primaryfragments in Fig 2, it is evident that the isoscaling is valid for a limited range of isotopes forthe secondary fragments as compared to the primary ones. When the isotopes away fromthe valley of stable nuclei are considered, the trends for the secondary fragments are not asclearly consistent with the isoscaling law as are the trends of the primary distribution. Onecan conclude from this that isoscaling is approximately valid in the case of the secondaryfragments. The lines in the figures are the best fits of the calculated R ratios which agree9losely with the experimental data. The temperature required to reproduce isoscaling datawith the primary fragments is about 8 MeV[7] whereas that required for the secondaryfragments is 5.8 MeV. This decrease was already predicted in one of our earlier papers[7]in Sec.9. It has also been found out by Ono et al.[8] that the effect of secondary decay isto decrease the isoscaling parameter α by about 50%. This is indeed what emerges fromour calculations after including secondary decay code with the canonical model though theamount of reduction is less as compared to the dynamical model. Tha value of α as obtainedfrom the fits of the primary fragments (left panel of Fig. 2) is 0.713 whereas that obtainedfrom the secondary fragments is 0.580 which is much closer to the experimentally obtainedvalue for α [17] equal to 0.566.In Fig. 4 we have also plotted the isoscaling ratios for different neutron number valuesas function of the proton number Z and thereby calculated the other isoscaling parameter β from them. The value of β as obatined from the linear fits of the primary fragments( rightpanel of Fig. 2) is -0.849. For the secondary fragments, the value of β from our model is-0.634 whereas the experimentally obtained value is -0.621. The value of β also decreases inthe case of secondary from the primary ones as in the case of α . The results obtained afterthe sequential decay matches very well with the experimental ratios. As for the isotopes inFig 3, for the isotones also it is seen that the linear isoscaling is valid for a limited range incase of secondary fragments as compared to the primary ones(right panel of Fig.2).We now turn to the case where the target is T a . One can consider the extreme limitwhich is target independence and by which we mean that N , Z refers to simply to thecase where just the projectile is the disintegrating system. The more likely scenario wheresome matter has diffused to or from the target has many possibilities. In principle, thetarget could shear away some material from the projectile leaving a PLF which is a fractionof the projectile. For a peripheral collision this is less likely than the alternative of theprojectile picking up part of nuclear matter from the tail region of the much larger target.The amount of nuclear matter curved out of Ta will be small(for the disintegrating system toretain PLF characteristics) but other than that not much can be said and an integration overthe different possibilities might be essential. Keeping this limitation in mind, we comparedwith different possible scenarios and the case with projectile plus 10 neutrons and 8 protonsfrom the target, i.e, projectile + T a instead of Be . The theoretical slope α as obtained from Fig.5 from the fit of the secondary fragments is 0.459 while the experimentally obtained slopeis 0.432. The straight line fit to the calculated points matches nicely with the experimentalratios. The value of β as obtained from our model from the slopes in Fig. 6 is -0.489 whereasthe experimentally obtained value is -0.487. Thus we find that the values of the isoscalingparametrs as obtained from the secondary fragments for both the targets agree quite wellwith the experimental values as can be seen from Table 1. Target material Isoscaling parameters primary secondary experiment Be α ( Z min = 6 , Z max = 13) 0.713 0.580 0.566 Be β ( N min = 8 , N max = 15) -0.849 -0.634 -0.621 T a α ( Z min = 6 , Z max = 13) 0.619 0.459 0.432 T a β N min = 8 , N max = 15) -0.682 -0.489 -0.487TABLE I: Best fit values of the isoscaling parametrs α and β for the two targets Be and T a . Thevalues obtained from the slope of the primary and secondary fragments as well as the experimentalvalues are tabulated. In the second column the range of Z or N values used to calculate theparameters are indicated. IV. SUMMARY
This work deals with the developing of the sequential decay code and successfully couplingit with the canonical thermodynamical model in order to compare the properties of thesecondary fragments with the experimental data. The width, peak position and rapid fallin cross-section of the isotopic distribution of the secondary fragments matches well withthe experimental data. The main purpose is to examine the effects of sequential decayon the phenomenon of isoscaling. The secondary fragments also shows isoscsling and thethe isoscaling parameters calculated from the secondary fragments matches closely with theexperimental data. The temperature required to reproduce the experimental data with the11econdary fragments is less than needed by the primary ones. We finally conclude thatthe canonical thermodynamical model can explain the isoscaling properties of the lighterfragments produced in the projectile fragmentation reaction.
V. ACKNOWLEDGEMENT
The authors gratefully acknowledges important discussions with Prof. Subal Dasgupta.They are thankful to Prof. M.B. Tsang and Dr. M. Mocko. Valuable suggestions from Dr.Santanu Pal and Jhilam Sadhukhan is also acknowledged gratefully. [1] M. Mocko et al., Phys. Rev C , 054612 (2006)[2] C. B. Das, S. Das Gupta, W. G. Lynch, A. Z. Mekjian, and M. B. Tsang, Phys. Rep , 1,(2005).[3] G. Chaudhuri, S. Das Gupta, W. G. Lynch, M. Mocko and M.B. Tsang, Phys. Rev. C , 716, (2000).[5] M. B. Tsang, W. A. Friedman, C. K. Gelbke, W. G. Lynch, G. Verde, and H. S. Xu, Phys.Rev. Lett. , 5023 (2001).[6] M. B. Tsang et al., Phys. Rev C , 054615 (2001).[7] G. Chaudhuri, S. Das Gupta and M. Mocko, Nucl. Phys. A , 293-313 (2008).[8] A. Ono et al.,arxiv:nucl-ex/0507018v2.[9] Y. Fu, D.Q. Fang, Y.G. Ma, X.Z. Cai, X.Y. Sun and W.D. Tian, Nucl. Phys. A , 584c(2010).[10] K.C. Chase and A. Z. Mekjian, Phys. Rev. C , R2339(1995).[11] J. P. Bondorf, A. S. Botvina, A. S. Iljinov, I. N. Mishustin and K. Sneppen, Phys. Rep. ,133 (1995).[12] V. Weiskoff, Phys. Rev. , 295(1937).[13] W.A. Friedman and W. G. Lynch, Phys. Rev C , 16 (1983).[14] J. E. Lynn, Theory of Neutron Resonance Reactions ,Clarendon, Oxford, 1968, p-325.
15] C. Guaraldo, V. Lucherini, E. D. Sanctis, A.S. Iljinov, M. V. Mebel and S. Lo Nigro, NuovoCimento , 4 (1990).[16] G. Chaudhuri, PhD thesis(Chapter IV), arXiv:nucl-th/0411005[17] M. Mocko, PhD thesis, Michigan State University 2006. -6 -5 -4 -3 -2 -1 A Mg Ni +Be -6 -5 -4 -3 -2 -1 A Mg Ni +Be C r o ss - s e c t i on ( m b ) Neutron Number(N)
FIG. 1: Experimental cross sections of magnesium isotopes(squares with error bars) comparedwith theoretical results: primary fragments( open stars joined by dotted line) and secondary frag-ments(open triangles joined by solid line). The left panel is for the reaction Ni on Be while theright panel is for Ni on Be reaction. The temperature is 5.8 MeV for both the reactions. -3 -2 -1 Z=7 Z=13Z=12Z=11Z=10Z=6Z=9Z=8 R ( N , Z ) Neutron Number(N) -3 -2 -1 Ni on Be T=5.8 MeV
N=15N=14N=13N=12N=11N=10N=9N=8
Proton Number(Z)
FIG. 2: Ratios( R ) of multiplicities of primary fragments of producing the nucleus ( N, Z ) wherereaction 1 is Ni on Be and reaction 2 is Ni on Be. The left panel shows the ratios as functionof neutron number N for fixed Z values from 6 to 13, while the right panel displays the ratios asfunction of proton number Z for fixed neutron numbers from N = 8 to 15. The lines drawn throughthe theoretical points(open triangles) are best fits of the calculated ratios. The temperature usedfor both the reacions is 5.8 MeV. -1 R ( N , Z ) Z=12Z=10Z=8Z=6 -1 Ni on Be T=5.8 MeV
Z=13Z=11Z=9Z=7
Neutron Number(N)
FIG. 3: Ratio of multiplicities of the secondary fragments of producing the nucleus (
N, Z ) wherereaction 1 is Ni on Be and reaction 2 is Ni on Be compred with the ratios of the experimentalcross sections of the same two reactions. The left panel shows the even Z isotopes while the rightpanel shows the results for the odd ones. The lines drawn through the theoretical points(opentriangles) are best fits of the calculated ratios. The experimental points are shown by solid squareswith error bars.The temperature used for both the reacions is 5.8 MeV. -1 -1 Ni on Be T=5.8 MeV
N=14N=12N=10 N=9 N=15N=13N=11N=8 R ( N , Z ) Proton Number(Z)
FIG. 4: Same as Fig. 3 except that the ratios are plotted as function of the proton number Z forfixed neutron numbers. The left panel shows the results for the even neutron numbers while theright ones show those for the odd ones. -1 Ni on Ta T=6.2 MeV R ( N , Z ) Neutron Number(N)
Z=12Z=10Z=8Z=6 -1 Z=13Z=11Z=9Z=7
FIG. 5: Same as Fig. 3 except that here the target is
T a instead of Be . The temperatureused for both the the reactions is 6.2 MeV. -1 R ( N , Z ) N=14N=12
N=10N=8 -1 Proton Number(Z) Ni on Ta T=6.2 MeV
N=9 N=11 N=13 N=15
FIG. 6: Same as Fig. 4 except that here the target is
T a instead of Be . The temperatureused for both the reactions is 6.2 MeV.. The temperatureused for both the reactions is 6.2 MeV.