Projectile Fragmentation of 86 Kr at 64 MeV/nucleon
M. Mocko, M. B. Tsang, Z. Y. Sun, N. Aoi, J. Cook, F. Delaunay, M. A. Famiano, H. Hui, N. Imai, H. Iwasaki, W. G. Lynch, T. Motobayashi, M. Niikura, T. Onishi, A. M. Rogers, H. Sakurai, A. Stolz, H. Suzuki, E. Takeshita, S. Takeuchi, M. S. Wallace
PProjectile Fragmentation of Kr at 64 MeV/nucleon
M. Mocko,
1, 2, ∗ M. B. Tsang,
1, 2
Z. Y. Sun, N. Aoi, J. Cook,
1, 2
F. Delaunay, M. A. Famiano, H. Hui, N. Imai, H. Iwasaki, W. G. Lynch,
1, 2
T. Motobayashi, M. Niikura, T. Onishi, A. M. Rogers,
1, 2
H. Sakurai, A. Stolz, H. Suzuki, E. Takeshita, S. Takeuchi, and M. S. Wallace
1, 2 National Superconducting Cyclotron Laboratory, MichiganState University, East Lansing, Michigan 48824, USA Department of Physics & Astronomy, MichiganState University, East Lansing, Michigan 48824, USA Institute of Modern Physics, CAS, Lanzhou 730000, China RIKEN Nishina Center, 2-1 Hirosawa, Wako, Saitama 351-0198 Japan Department of Physics, University of Tokyo, 7-3-1 Hongo, Bunkyo, Tokyo 113-0033, Japan Center for Nuclear Study, University of Tokyo (CNS), RIKENcampus, 2-1, Hirosawa, Wako, Saitama 351-0198, Japan Rikkyo University, 3 Nishi-Ikebukuro, Toshima, Tokyo 171, Japan (Dated: October 26, 2018) a r X i v : . [ nu c l - e x ] M a y bstract We measured fragmentation cross sections produced using the primary beam of Kr at 64MeV/nucleon on Be and
Ta targets. The cross sections were obtained by integrating themomentum distributions of isotopes with 25 ≤ Z ≤
36 measured using the RIPS fragment separatorat RIKEN. The cross-section ratios obtained with the
Ta and Be targets depend on the fragmentmasses, contrary to the simple geometrical models. We compared the extracted cross sections toEPAX; an empirical parameterization of fragmentation cross sections. Predictions from currentEPAX parameterization severely overestimate the production cross sections of very neutron-richisotopes. Attempts to obtain another set of EPAX parameters specific to the reaction studied here,to extrapolate the neutron-rich nuclei more accurately have not been very successful, suggestingthat accurate predictions of production cross sections of nuclei far from the valley of stabilityrequire information of nuclear properties which are not present in EPAX.
PACS numbers: 25.70.MnKeywords: projectile fragmentation, fragmentation reactions, fragment separator, fragmentation productioncross section ∗ Corresponding author: [email protected] . INTRODUCTION With recent developments in heavy-ion accelerators and rare isotope beam productionmany new surprising phenomena have been observed in unstable nuclei, such as neutronhalo [1], neutron and proton skins of nuclei far from stability [2, 3], and large deformationsof neutron-rich isotopes [4]. In the planning and development of experiments with rareisotope beams, the EPAX code is used extensively in the current radioactive ion beamfacilities. EPAX is an empirical parameterization of fragmentation cross sections relying ondata mainly from reactions at incident energy greater than 200 MeV/nucleon. Using EPAXat low incident energy assumes the validity of limiting fragmentation, when the productioncross sections do not depend on incident energy or target. It is, therefore, very importantto verify EPAX predictions of production of rare isotopes at extreme proton and neutroncompositions, especially for facilities that produce radioactive ion beams at incident energieslower than 200 MeV/nucleon.The present study compares fragment production cross sections from the projectile frag-mentation of Kr at 64 MeV/nucleon to EPAX, an empirical parameterization of fragmen-tation cross sections. Kr is chosen as it is one of the most neutron-rich naturally occurringstable isotopes. Due to its noble gas chemical properties and that it can be easily ionized inan ion source, projectile fragmentation of Kr is widely used to produce neutron-rich rareisotopes.
II. EXPERIMENTAL SETUP
The fragmentation experiments were carried out at RIKEN Accelerator Research Facility[5]. A primary beam of Kr with incident energy of 64 MeV/nucleon was produced byinjecting Kr ions into the K540 Ring Cyclotron using the LINAC injector. The layoutof the LINAC, K540 Ring Cyclotron, and the experimental areas in the RIKEN facilityis shown in Fig. 1. Two reaction targets, 96 mg/cm
Be and 156 mg/cm
Ta foils,were used. The target thicknesses were chosen such that the energy losses of the primarybeam in the targets were similar thus data could be taken with both targets using the samemagnetic setting. Minimizing the number of settings required in the experiments results inbetter utilization of the primary beam since changing the magnetic setting of the RIKEN3rojectile Fragment Separator (RIPS) takes much longer than changing the targets.Projectile-like fragments produced in interactions of the primary beam with the targetnuclei were collected and identified using the RIPS separator [6] located in experimentalareas D and E6 as shown in Fig. 1. The schematic layout of RIPS is shown in Fig. 2.The RIPS fragment separator consists of two 45 ◦ dipole magnets (D1, D2), and twelvequadrupoles (Q1–Q12). The first section gives a dispersive focus at the F1 focal plane al-lowing measurement of the magnetic rigidity of the particles. The second stage compensatesthe dispersion of the first section and gives a double achromatic focus at the F2 focal plane.The quadrupole triplet of the last section produces the third focus at the F3 focal plane,where the main part of the particle identification setup was installed.All measurements were performed using the RIPS fragment separator in a narrow mo-mentum acceptance mode. The momentum opening, d p/p , was limited to 0.2% using a slitin the dispersive image of the separator, F1 (see the top right oval in Fig. 2). In this config-uration, the measured particles have trajectories close to the axis of the fragment separatorsimplifying the transmission calculations. Furthermore, a narrow momentum acceptance al-lows measuring the fragment cross sections in the magnetic rigidity between primary beamcharge states. The disadvantage is that in order to measure the momentum distributionsover a wide range of fragmentation products, we had to take measurements at many dif-ferent magnetic settings. For reactions with the Be target we covered 1.79–2.93 Tm in45 steps and for
Ta target we scanned the region of 1.79–2.35 Tm in 29 settings. Toavoid excessive dead-time in the data acquisition the primary beam intensity was optimizedat each magnetic rigidity such that the counting rate of the first silicon PIN detector wasapproximately 900–1000 counts per second.Fragments with mass number, A , proton number, Z , and charge state, Q , measured inour study (25 ≤ Z ≤
36) were not fully stripped of electrons. However, only the charge statedistributions of the Kr primary beam were measured. The measurement was done at theF1 dispersion plane where different charge states of one ion traveling at the same velocityare spatially separated [7]. The measured primary charge state probability distributions for Be (filled circles) and
Ta (filled squares) targets are plotted in Fig. 3 as a function ofthe number of unstripped electrons, Z − Q . Predictions from the charge state distributioncode GLOBAL [8], as implemented in LISE++ [9], are shown as solid and dotted lines for Be and
Ta targets. The predictions tend to decrease more steeply for the Kr+ Be4eactions. The overall prediction is quite good considering the fact that the GLOBAL codewas developed for heavier projectiles (
Z >
53) at higher energies (
E >
100 MeV/nucleon)[8]. The measured charge state distribution of the Kr primary beam showed that almost10% of the intensity is in the Kr charge state after passing through the Be target(Fig. 3). The fraction is much larger in the case of
Ta target because the charge statedistribution is broader.To properly identify all fragments and their charge states in our analysis, the general Bρ - T oF -∆ E - T KE [10] particle identification technique was used on an event-by-event basis.The magnetic rigidity, Bρ , was given by the magnetic setting of the RIPS fragment separator.The time of flight, T oF , was measured between F2 and F3 plastic scintillators (see Fig. 2)separated by a flight path of 6 m. The energy loss, ∆ E , was measured with a 350 µ m-thicksilicon PIN detector. The total kinetic energy, T KE , was reconstructed by measuring theenergy deposited by the particles in a stack of 5 silicon PIN detectors (labeled ∆ E , E E E E E versus T oF , is shownin the left panel of Fig. 4. The identification of individual groups of events was done byrecognizing typical features of the PID spectrum and locating a hole corresponding to theparticle-unbound Be nucleus [11]. The spectrum for the Kr+ Be reaction at Bρ = 2 . Z = 28, 31, and 34. Theright panels display projections of events from these gates to charge state, Q , versus ratio A/Q plane. The fully stripped ( Z − Q = 0) and hydrogen-like ( Z − Q = 1) charge statesfor all 3 selected elements are very well separated. Similar projections were constructed forfragments with 25 ≤ Z ≤
36 at all magnetic rigidity settings in our analysis.Each experimental run took data for one Bρ setting of the RIPS fragment separator. Thenumber of events, N ( A, Z, Q ), for a fragment with mass number, A , proton number, Z , andcharge state, Q , were extracted from the calibrated PID spectra similar to the one in Fig.4. The differential cross sections, d σ/ d p , were calculated taking into account the numberof beam particles, N B , number of target nuclei per square centimeter, N T , live-time ratio, τ LIV E , and the transmission efficiency through the RIPS fragment separator, ε ,d σ d p ( A, Z, Q ) = N ( A, Z, Q ) N T N B ∆ pτ LIV E ε , (1)where ∆ p denotes the momentum opening. 5he transmission efficiency correction, ε , is assumed to be factorized into two independentcomponents: momentum corrections and angular corrections. Momentum corrections takeinto account the loss of fragments caused by the momentum slit at the F1 focal plane. Thiseffect is independent of fragment species and the Bρ setting. A correction value of 98 ± A , bya Gaussian distribution with variance, σ ⊥ , prescribed in ref. [13]: σ ⊥ = σ A ( A P − A ) A P − σ D A ( A − A P ( A P − , (2)where A P is the mass number of the projectile and σ D is the orbital dispersion. The first termin Eq. (2) comes from the Goldhaber model [14], which describes the width of longitudinalmomentum distribution of fragments produced at high projectile energies. The value of σ was determined by fitting the experimental longitudinal distributions. Values of 147 ± ± c were obtained for reactions with Be and
Ta targets, respectively.The second term in Eq. (2) takes into account the deflection of the projectile by the targetnucleus [15] and is significant only for fragments with masses close to the projectile andat low and intermediate beam energies. We estimated the σ D parameter to be 225 ± c for both investigated reactions, based on the O fragmentation data measured at 90MeV/nucleon [13]. Portions of the Gaussian angular distributions transmitted through theRIPS fragment separator define the angular transmission and were calculated using LISE++[9] and verified with MOCADI simulations [7]. The transmission correction, ε , consisting ofthe product of the angular and momentum corrections is plotted in Fig. 5 for the Kr+ Bereaction. The final transmission correction, ε , varies from 0.98 for fragments close to theprojectile to approximately 0.25 for the lightest fragments in our analysis ( A ≈ Kr+
Ta reaction is very similar to the one shown in Fig.5. In our fragmentation measurements the beam intensity varied between 10 and 10 pps. The beam intensity was monitored by a telescope located at approximately 60 ◦ withrespect to the beam direction and approximately 25 cm from the target. The top left oval6n Fig. 2 shows a schematic drawing of the monitor (MOMOTA) at the target position.The monitor consists of three plastic scintillators and detects the light particles produced innuclear reactions in the production target. Only triple coincidence rates were considered asvalid signals. Since the production of light particles depends on the reaction of beam andtarget nuclei, the monitor rates must be calibrated to the beam intensity for each reactionsystem studied. Unfortunately, we could not use the Faraday Cup (FC) to calibrate thebeam intensity. The FC was located approximately 5 cm downstream from the targetposition and the monitor reading was affected by the particles scattered off the FC duringthe primary beam intensity calibration. To obtain an absolute calibration of the monitor,direct rates of Kr and Kr particles for the Be and
Ta targets, respectively, weremeasured at the F2 focal plane using the plastic scintillator. The statistical uncertainties ofthese measurements were less than 5%. From Fig. 3, probabilities of Kr and Kr charge states are found to be 0.0028% and 0.016%, respectively. This allowed us to calculatethe primary beam intensity for these two measurements, thus establishing absolute beamintensity calibration points for the Be and
Ta targets. The linearity (better than 1%) inthe beam intensity range used in our experiments for the monitor telescope was confirmedby measuring the fragment flux with different F1 slit openings.
III. MOMENTUM DISTRIBUTIONS
The fragment momentum distributions were obtained by plotting individual differentialcross sections as a function of measured momentum (calculated from the magnetic rigidity, Bρ ) for all fragments and their charge states. The momentum distributions obtained fromprojectile fragmentation at intermediate energy are asymmetric [7, 10]. Fig. 6 displays atypical momentum distribution in our analysis for Zn . The dashed curve represents afit with a single Gaussian function. As the distributions have low momentum tails, we fitthe data with the following function [7, 16]: dσdp = S · exp ( − ( p − p ) / (2 σ L )) for p ≤ p ,S · exp ( − ( p − p ) / (2 σ R )) for p > p , (3)where S is the normalization factor, p , is the peak position of the distribution, and σ L and σ R are widths of “left” and “right” halves of two Gaussian distributions used to fitthe momentum distributions. The solid curves in Fig. 6 are the best fits obtained by7inimization of χ using Eq. (3). For most fragments we observe very good agreementbetween the data and the fit over three orders of magnitude. IV. CROSS-SECTION MEASUREMENTS
The cross section of a fragment in a given charge state was determined by integrating thearea of its momentum distribution. For fragments with well-measured momentum distribu-tions, such as the one shown in Fig. 6, the cross sections were extracted from fitting themomentum distributions using Eq. (3). However, approximately 40% of the measured frag-ments had incomplete momentum distributions that may consist of only a few points nearthe top of the peak. For these fragments, we used the systematics of p , σ L , and σ R obtainedfrom fragments with complete momentum distributions to calculate the cross sections withfunction in Eq. (3).At 64 MeV/nucleon, the fragment yield is distributed over different charge states. Thetotal fragmentation cross sections are obtained by summing these contributions. For the Kr+ Be reaction system we analyzed fully stripped fragments with Z − Q = 0 charge statesand corrected the final fragment cross sections using charge state distributions predicted byGLOBAL. The calculated corrections vary between 1–9% for 25 ≤ Z ≤
36 isotopes. Forthe Kr+
Ta reaction we sum the cross sections of the 3 most abundant charge states( Z − Q = 0, 1, 2) to harvest most of the cross section. Corrections for fragment crosssections using GLOBAL vary between 0.1–3% for 25 ≤ Z ≤
36 isotopes.For fragments with complete momentum distributions, uncertainties in the fragmentationcross sections of 7–12%, were calculated based on the statistical uncertainty, the beam inten-sity calibration (5%), the errors from the fitting procedure and the transmission uncertainty(2–8%). For fragments measured with incomplete momentum distributions, additional sys-tematic errors stemming from the extrapolation of the parameters of p , σ L , and σ R wereincluded. An overall view of the fragment cross sections for the Kr+ Be reaction system inthe style of the nuclear chart, is shown in Fig. 7. The range of the measured cross sectionsspans over 9 orders of magnitude, from 15 ± Cu) to 38 ± Kr).8 . CROSS-SECTION RESULTS
Fig. 8 shows the cross sections for fragments extracted from the Kr+
Ta analysis asclosed circles. Each panel represents isotope cross-section data for one element (25 ≤ Z ≤ N − Z , of each isotope. For the Kr+
Tareaction system, interference from the many charge states of the beam limits the span ofmeasured fragments for each element. Our requirement, that the three most abundant chargestates should have quantifiable counts above background in the analysis further reduced thenumber of data points to 70 isotopes for the Kr+
Ta system. In contrast, cross sectionsfor 180 isotopes were obtained for the Kr+ Be system as shown in Fig. 9.For comparison, fragment cross sections for the Kr+ Be reactions are plotted as opensquares in Fig. 8. More light fragments are produced in the projectile fragmentation of the Kr nuclei with
Ta than Be targets. This increase is seen clearly in Fig. 10 where theratios of isotope yields from the two different targets, σ Ta ( A, Z ) /σ Be ( A, Z ), are plotted asa function of fragment mass number, A , and σ Ta ( A, Z ) and σ Be ( A, Z ) denote cross sectionsof an isotope (
A, Z ) measured with
Ta and Be targets, respectively. For clarity of thepresentation, only the target isotope ratios with relative errors smaller than 25% are shown.Elements with odd and even Z s are represented by open and closed symbols, respectively,with the open circles starting at A ≈
52 representing the Mn isotopes and the solid trianglesnear A ≈
80 denoting the Kr isotopes. Within an element (data points with same symbol),there seems to be an increase in the fragment cross sections from reactions with Ta targetsfor both very neutron-rich and proton-rich isotopes. The trend is not as clear here dueto the limited range of isotopes measured in the Kr+
Ta reactions. (Similar trendshave been observed in the projectile fragmentation of
Ca and
Ni isotopes [7].) Theexperimental target isotope ratios, σ Ta ( A, Z ) /σ Be ( A, Z ), exhibit an overall increase withdecreasing fragment mass in Fig. 10. For fragments lighter than A ≈
50, the enhancementexceeds a factor of 10. Such dependence is not expected in the limiting fragmentationmodel. In the geometrical limit the cross sections are proportional to the sum of nuclearradii squared [17], so the target isotope ratios are given by: σ Ta ( A, Z ) σ Be ( A, Z ) = (cid:16) A / + A / (cid:17) (cid:16) A / + A / (cid:17) = 2 . , (4)where A Kr = 86, A Ta = 181, and A Be = 9. This limit is shown as a dotted line in the figure.9n the EPAX formula the fragmentation cross section is proportional to the sum of nuclearradii, which stems from the assumption that fragmentation is dominated by peripheralevents: σ Ta ( A, Z ) σ Be ( A, Z ) = (cid:16) A / + A / − . (cid:17)(cid:16) A / + A / − . (cid:17) = 1 . . (5)This EPAX limit is shown as a dashed line in the figure. The cross-section enhancementtrends suggest that light, rare isotopes may be produced more abundantly using a heavytarget such as Ta. However, one must keep in mind the large difference in atomic mass ofthe two target materials (approximately a factor of 20). To compensate for the low atomicdensity in Ta or similar targets, thick foils must be used, and effects such as the broad chargestate distribution for heavy targets, the energy loss, and angular straggling must be takeninto account. However, if the rising trend of the target isotope ratios for the Kr primarybeam continues for light isotopes, heavy targets such as Ta may be a better choice for theproduction of light neutron-rich and proton-rich isotopes close to the drip lines [18].For both investigated systems, we also observed differences between the EPAX calculatedand observed maxima of the isotopic distribution for elements close to the projectile (Ge–Kr). A similar systematic discrepancy between the intermediate energy fragmentation dataand EPAX parameterization has been reported before [7, 19]. The Fermi spheres of thetarget and projectile nuclei have larger overlap at intermediate energies than at relativisticenergies. There may be increasing contributions to the prefragments with charge numbersgreater than that of the projectile from the transfer-type reactions. Subsequent decay ofthese primary fragments feeds the less neutron-rich isotopes close to the projectile.The parameters used in EPAX were obtained by fitting several data sets, including thefragmentation data of Kr+ Be at 500 MeV/nucleon [20]. For comparison, the latter setof data was plotted as open triangles in Fig. 9, and our data are plotted as closed squares.There are considerable scatters in the Weber et al. data (especially for Ga to Se elements).The cross sections at the peak of the isotopic distributions for Co to Zn elements agree ratherwell. However, the 500-MeV/nucleon isotope distributions are wider. These may account forthe larger widths from the calculated isotope distributions by EPAX. It has been known thatEPAX over-predicts the production of very neutron-rich fragments [11, 16]. The top panelof Fig. 11 shows the ratio of the measured cross sections divided by the EPAX predictionsas a function of the neutron number from the neutron stability line, N β . For convenience,10e adopt the same stability line for a chain of isobars, A , as used in EPAX [21]: N β = A − A . − . A / . (6)Other choices of the stability line lead to the same conclusions. The same convention ofthe symbols used in Fig. 10 is adopted here with the open circles (top left corner in Fig.10) denoting Z = 25 isotopes and closed triangles (lower right corner in Fig. 10) denote Z = 36 isotopes. EPAX predicts isotopes near the stability line to better than a factor of 2.However, starting around two neutrons beyond the EPAX stability line, over-prediction fromEPAX worsens with increasing neutron richness for a fixed element. By extrapolating theproton-removed isotopes ( N = 50) from the Kr projectiles (the right-most points joinedby the dashed curve), the over-prediction of the rare neutron-rich nuclei such as Ni couldbe a factor of 100.To examine the behavior of EPAX predictions with respect to neutron-rich nuclei, we plotthe ratios of σ ( Kr+ Be) /σ EPAX as a function of the atomic number of the fragments for 42 ≤ N ≤
50 isotones in Fig. 12. The open circles represent predictions from the standard EPAXcalculations. In each panel, the neutron-rich isotopes are those with lowest Z . Aside from thepick-up reactions, the most neutron-rich fragments created in the projectile fragmentationreactions of Kr are isotones with N = 50 (lower right panel). In most cases, the lastdata point with lowest Z in each isotone chain is only a couple proton numbers away fromthe most neutron-rich known nuclei. Thus, EPAX predictions on the production of veryproton rich and neutron rich isotopes can be off by more than an order of magnitude.Since neutron-rich nuclei are of interest to a variety of problems in astrophysics and nuclearstructure the demand for such beams is high. Unfortunately, the inaccuracy in the beamrate estimation using EPAX presents large uncertainties in designing experiments involvingthese rare isotopes.Since the EPAX parameters result from fitting the projectile fragmentation data of Ar, Ca, Ni, Kr,
Xe, and
Pb with the beam energy above 200-MeV/nucleon heavy-iondata, better fitting parameters may be obtained if only the present data set is used. Thenew set of fitting parameters may allow more accurate extrapolation to the yields of veryneutron rich nuclei.In the original version of EPAX, as briefly described in Appendix A, a total of 24 fittingparameters was obtained. Table I lists the parameters used in the original EPAX as well as11he modified EPAX parameters used to fit the present data. (For convenience, we label theEPAX calculations using the new set of parameters EPAX Kr .) The bottom panel of Fig.11 shows the ratio of data over the predictions from EPAX Kr . Compared to the top panel,the overall agreement with the experimental data is much better. This is not surprisingconsidering EPAX Kr is not a global fit and describes the cross sections for only one reaction.To study how the extrapolations would behave in the neutron-rich region, the new ratios ofdata over the predictions of EPAX Kr are plotted as closed points in Fig. 12. Contrary to theratios using original EPAX parameters, the new ratios are less than a factor of two over alarge Z range. However, the behavior of the most neutron rich nuclei ratios do not exhibit apredictable dependence on Z . Thus accurate extrapolation to the unmeasured neutron-richregion (the left side of each panel with smaller Z for fixed N ) cannot be obtained. This couldbe due to the fact that EPAX is a fitting code that does not include the properties of exoticnuclei such as binding energy or neutron separation energy [22] . Better extrapolations willrequire the use of models that include more physics. However, discussions of such modelsare beyond the scope of this paper. VI. SUMMARY
Fragmentation production cross sections have been measured for Kr primary beam on Be and
Ta reaction targets at 64 MeV/nucleon. The cross-section ratios obtained withthe
Ta and Be targets show a fragment mass and charge dependence, contrary to thesimple geometrical models. The isotopic distributions of fragments produced in Kr+ Bereactions are narrower than those calculated by the EPAX formula resulting in severe cross-section over-predictions for the very neutron-rich isotopes. The availability of comprehensivedata, such as those presented here, suggests that it is difficult to extrapolate accurately thecross sections of exotic neutron-rich nuclei with different EPAX fitting parameters [16, 23,24]. Away from the stability, properties of the exotic nuclei become important, and EPAXdoes not include basic nuclear property information such as the binding energy.12 cknowledgments
We would like to thank the operation group of Riken for producing high quality andhigh intensity Kr beam during our experiment. We thank Dr. K. S¨ummerer for giving usinvaluable insights on fitting the EPAX parameters. This work is supported by the NationalScience Foundation under Grant Nos. PHY-01-10253, PHY-0606007 INT-0218329, andOISE*-0089581.
APPENDIX: EPAX PARAMETERIZATION
In the EPAX parameterization [21] the fragmentation cross section of a fragment withmass, A , and nuclear charge, Z , created from projectile ( A p , Z p ) colliding with a target ( A t , Z t ) is given by: σ ( A, Z ) = Y A n exp (cid:16) − R | Z prob − Z | U n(p) (cid:17) . (A.1)The first term Y A describes the sum of the isobaric cross sections with A . The second term,exp (cid:16) − R | Z prob − Z | U n(p) (cid:17) , is called the “charge dispersion,” and describes the distribution ofthe elemental cross sections around the maximum value, Z prob , for a given mass. The shapeof the charge distribution is controlled by the width parameter, R , and the exponents, U n and U p , describe the neutron-rich (n) and proton-rich (p) side, respectively. The neutron-rich fragments are defined with Z prob − Z > n = (cid:113) R/π normalizes the integral of the charge dispersion to unity.The mass yield, Y A , is parameterized as an exponential function of the number of removednucleons, A p − A : Y A = SP exp [ − P ( A p − A )] . (A.2) S is the overall scaling factor that accounts for the peripheral nature of the fragmentationreaction and proportional to the sum of the projectile and the target radii: S = S ( A / p + A / t + S ) . (A.3)with S and S being fitting parameters.The slope of the exponential function in Equation (A.2), P , is taken as a function of theprojectile mass, A p , with P and P as fitting parameters: P = exp ( P A p + P ) . (A.4)13he charge dispersion term, exp (cid:16) − R | Z prob − Z | U n(p) (cid:17) , in Equation (A.1) is described bythree parameters R , Z prob , and U n ( p ) . These parameters are strongly correlated [21].The width parameter, R , of the charge distribution is parameterized as a function of thefragment mass, A , with R and R as fitting parameters: R = exp ( R A + R ) . (A.5)To account for the asymmetric nature of the shape of isobaric distributions, the exponents, U n and U p , for the neutron-rich and proton-rich sides are different. U n = U n + U n A (A.6) U p = U + U A + U A (A.7)The maximum of the isobar distribution, Z prob , lies in the valley of stability and it isparameterized as: Z prob ( A ) = Z β ( A ) + ∆ , (A.8)where Z β ( A ) is approximated by a smooth function of the mass number, A : Z β ( A ) = A .
98 + 0 . A / , (A.9)and the ∆ parameter is found to be a linear function of the fragment mass, A , for heavyfragments and a quadratic function of A for lower masses:∆ = ∆ A + ∆ if A ≥ ∆ , ∆ A if A < ∆ , (A.10)where ∆ , ∆ , ∆ , and ∆ are EPAX parameters.The above description from Eq. (A.1) to (A.10) is sufficient to predict the cross sectionsof fragments located close to the line of stability and far from the projectile nucleus, alsoreferred to as the “residue corridor.” For fragments with masses close to the projectile,corrections to the parameters ∆, R , and Y A are introduced, according to the followingequations: ∆ = ∆ (cid:104) d ( A/A p − d ) (cid:105) , (A.11) R = R (cid:104) r ( A/A p − r ) (cid:105) , (A.12) Y A = Y A (cid:104) y ( A/A p − y ) (cid:105) , (A.13)14or ( A/A p − d ) >
0, (
A/A p − r ) >
0, and (
A/A p − y ) >
0, respectively.A final correction is applied in the case of projectile nuclei far from the line of β -stability, Z β ( A p ). In this case, the fragment distributions keep some memory of the A/Z ratio of theprojectile nucleus resulting in a correction to the maximum, Z prob , of the charge distribution: Z prob ( A ) = Z β ( A ) + ∆ + ∆ m , (A.14)where ∆ m is expressed separately for neutron-rich (( Z p − Z β ( A p )) <
0) and proton-rich(( Z p − Z β ( A p )) >
0) projectiles:∆ m = ( Z p − Z β ( A p )) [ n ( A/A p ) + n ( A/A p ) ] for neutron rich , ( Z p − Z β ( A p )) exp [ p + p ( A/A p )] for proton rich , (A.15)where n , n and p , p are fitting parameters.The EPAX parameterization altogether contains 24 parameters ( S , S , P , P , R , R ,∆ , ∆ , ∆ , ∆ , U n , U n , U , U , U , n , n , p , p , d , d , r , r , y , and y ), many of whichare strongly intercorrelated. The values used are listed in the middle column in Table I.The present set of data of Kr+ Be does not have as extensive mass range as the datafrom Ref. [20]. Therefore, Eq. (A.10) is reduced to fitting only one mass region with oneparameter, ∆ . Similarly, we do not make corrections to ∆ in Eq. (A.11). We also foundsome improvement if Eq. (A.7) is mass dependent. (The parameter U n in that equationwas absent in the original EPAX fitting.) All the parameters used in EPAX Kr are listed inthe rightmost column in Table I. Note that these are best-fit parameters to our data andcannot not be applied to other reactions or at different energies.15 ABLE I: Parameter values for EPAX [21] and EPAX Kr . EPAX Kr parameters are obtained byfitting the Kr+ Be reaction cross-sections.Parameter EPAX EPAX Kr S − .
38 0.0 S P − . − . P − . × − − . × − R R − . × − − . × − ∆ -1.087 N/A∆ . × − N/A∆ . × − . × − ∆ U n U n N/A 9 . × − U U . × − − . U − . × − . × − n − . n p − . − . p d − . d r − . r y − . y
1] I. Tanihata, Nucl. Phys. A , 275c (1991).[2] I. Tanihata, D. Hirata, T. Kobayashi, S. Shimoura, K. Sugimoto, and H. Toki, Phys. Lett. B , 261 (1992).[3] N. Fukunishi, T. Otsuka, and I. Tanihata, Phys. Rev. C , 1648 (1993).[4] T. Motobayashi, Y. Ikeda, Y. Ando, K. Ieki, M. Inoue, N. Iwasa, T. Kikuchi, M. Kurokawa,W. Moryia, S. Ogawa, et al., Phys. Lett. B , 9 (1995).[5] Y. Yano, in Proceedings 12th Int. Conf. on Cyclotrons and their applications , edited by B. Mar-tin and K. Ziegler (Word Scientific, 1989).[6] T. Kubo, M. Ishihara, N. Inabe, H. Kumagai, I. Tanihata, and K. Yoshida, Nucl. Instrum.Methods Phys. Res., Sect. B , 309 (1992).[7] M. Mocko, Ph.D. thesis, Michigan State University (2006).[8] C. Scheidenberg, T. St¨ohlker, W. E. Meyerhof, H. Geissel, P. H. Mokler, and B. Blank, Nucl.Instrum. Methods Phys. Res., Sect. B , 441 (1998).[9] D. Bazin, O. Tarasov, M. Lewitowicz, and O. Sorlin, Nucl. Instrum. Methods Phys. Res.,Sect. A , 307 (2002), URL .[10] D. Bazin, D. Guerreau, R. Anne, D. Guillemaud-Mueller, A. C. Mueller, and M. G. Saint-Laurent, Nucl. Phys. A , 349 (1990).[11] M. Mocko, M. B. Tsang, L. Andronenko, M. Andronenko, F. Delaunay, M. A. Famiano,T. Ginter, V. Henzl, D. Henzlov´a, H. Hua, et al., Phys. Rev. C , 054612 (2006).[12] N. Iwasa, H. Geissel, G. M¨unzenberg, C. Scheidenberger, T. Schwab, and H. Wollnik, Nucl.Instrum. Methods Phys. Res., Sect. B , 284 (1997).[13] K. V. Bibber, D. L. Hendrie, D. K. Scott, H. H. Weiman, L. S. Schroeder, J. V. Geaga, S. A.Cessin, R. Treuhaft, Y. J. Grossiord, J. O. Rasmussen, et al., Phys. Rev. Lett. , 840 (1979).[14] A. S. Goldhaber, Phys. Lett. B , 306 (1974).[15] R. Dayras, A. Pagano, J. Barrette, B. Berthier, D. M. D. C. Rizzo, E. Chavez, O. Cisse,R. Legrain, M. C. Mermaz, E. C. Pollacco, et al., Nucl. Phys. A , 299 (1986).[16] M. Notani, H. Sakurai, N. Aoi, H. Iwasaki, N. Fukuda, Z. Liu, K. Yoneda, H. Ogawa, T. Teran-ishi, T. Nakamura, et al., nucl-ex/0702050v1.[17] S. Kox, A. Gamp, P. Cherkaoui, A. J. Cole, N. Longequeue, J. Menet, C. Perrin, and J. B. FIG. 1: Layout of the experimental facility at RIKEN. The LINAC injector and the K540 cyclotronare shown along with the experimental areas E1–E6. The RIPS fragment separator is located inexperimental areas D and E6. [6]Viano, Nucl. Phys. A , 162 (1984).[18] H. Sakurai, S. M. Lukyanov, M. Notani, N. Aoi, D. Beaumel, N. Fukuda, M. Hirai, E. Ideguchi,N. Imai, M. Ishihara, et al., Phys. Lett. B , 180 (1999).[19] K. S¨ummerer, Nucl. Instrum. Methods Phys. Res., Sect. B , 278 (2003).[20] M. Weber, C. Donzaud, J. P. Dufour, H. Geissel, A. Grewe, D. Guillemaud-Mueller, H. Keller,M. Lewitowicz, A. Magel, A. C. Mueller, et al., Nucl. Phys. A , 659 (1994).[21] K. S¨ummerer and B. Blank, Phys. Rev. C , 034607 (2000).[22] W. A. Friedman and M. B. Tsang, Phys. Rev. C , 051601 (2003).[23] K. S¨ummerer, private communications.[24] Z. Y. Sun, private communications. F0 Q4 D1 Q1 Q7 D2 Q12
Momentum Slit
Primary Beam
Pdp = Kr F2 plastic (TOF start)F3 plastic (TOF stop)Si stackdE+E1+E2+E3+E4
FragmentsFragments
F1 F2F3
Production Target(Be,Ta)
MOMOTA
FIG. 2: RIPS fragment separator consisting of two dipoles (D1 and D2) and twelve quadrupoles(Q1–Q12). The momentum acceptance was determined by the momentum slit placed at F1. Thebeam intensity monitor (MOMOTA) is shown in the top left oval below the target position. Theparticle identification setup was located at the F2 and F3 focal planes.
Z-Q P r obab ili t y ( % ) -4 -3 -2 -1 Kr Be targetTa target
FIG. 3: Primary beam charge state distributions for Kr+ Be (closed circles) and Kr+
Ta(closed squares) plotted as a function of number of unstripped electrons, Z − Q . Solid and dashedcurves show calculation by GLOBAL code [8] as implemented in LISE++ [9] for Be and
Tatargets, respectively. oF (channels)
700 750 800 D E ( c hanne l s ) Z=34Z=31 Z=28
Ratio A/Q C ha r ge s t a t e Q FIG. 4: (Color online) Particle identification spectrum for the Kr+ Be reaction measured at a2.07 Tm magnetic rigidity setting. Left panel shows the PID with three gates around elementswith Z = 28, 31, 34. Right panel shows the corresponding projections to charge state, Q , versus A/Q ratio plane of events within from bottom to top, respectively. ragment mass number A
60 70 80 T r an s m i ss i on c o rr e c t i on Be Kr+ FIG. 5: Dependence of the transmission correction factor, ε , on fragment mass number, A, for the Kr+ Be reactions. p (MeV/c) M e V / c m b dp s d -6 -5 -4 -3 -2 p (MeV/c) M e V / c m b dp s d -6 -5 -4 -3 -2 Be Kr + FIG. 6: Momentum distributions for Zn produced in fragmentation of Kr on the Be target.The solid curve represents a fit with Eq. (3) and the dotted curve is a Gaussian fit, to the rightside of the momentum distribution, to show the asymmetry of the experimental distribution. nFe CoNi CuZn GaGe As Se Br Kr27 29 31 33 35 37 39 41 43 45 47 49 Neutrons P r o t o n s projectilestable
10 mb -6
10 mb
86 9
Kr+ Be
FIG. 7: (Color online) Measured cross sections for 180 fragments produced in the Kr+ Bereactions. Mn Fe Co Ni Cu Zn Ga Ge As
10 2 4 6 81 SeSe Br Kr Neutron excess N-Z C r o ss s e c t i on ( m b ) FIG. 8: Measured cross sections presented as isotope distributions for 25 ≤ Z ≤
36 elementsdetected in the Kr+
Ta reactions (filled circles) and in the Kr+ Be reactions (open squares)at 64 MeV/nucleon. EPAX calculations are shown as dashed ( Kr+ Be) and solid ( Kr+
Ta)curves. -2 Mn Fe Co -7 -4 -1 Ni Cu Zn -4 -1 Ga Ge As -3 -1
10 0 5 10 15 20 -3 -1 SeSe Br Kr Neutron excess N-Z C r o ss s e c t i on ( m b ) FIG. 9: Measured cross sections presented as isotope distributions for 25 ≤ Z ≤
36 elementsdetected in the Kr+ Be reactions at 64 MeV/nucleon. Experimental fragmentation data areshown as filled squares. EPAX predictions are shown as solid curves. For comparison, opentriangles show the published data of Kr+ Be at 500 MeV/nucleon [20]. ass number A
60 70 80 ( A , Z ) B e s ( A , Z ) / T a s Mass number A
60 70 80 ( A , Z ) B e s ( A , Z ) / T a s Kr Z=25 Z=36
FIG. 10: Ratios of the fragmentation cross sections on Ta and Be targets, σ Ta ( A, Z ) /σ Be ( A, Z ),for fragments with 25 ≤ Z ≤
36 for the Kr beam. Only ratios with relative errors smaller than25% are shown. Open and solid symbols represent odd and even elements starting with Z = 25.The horizontal dashed and dotted lines indicate the ratio calculated by the EPAX formula and Eq.(4), respectively. EPAX ) s / B e ) K r + ( s -1
84 83 82 81 8079Z=25Z=36 b N-N -2 0 2 4 6 ) K r ( EPAX s / B e ) K r + ( s -1 FIG. 11: Ratio of the experimental cross sections and predicted cross sections from EPAX (toppanel) and our modified EPAX Kr formula (bottom panel). For clarity, isotopes from each elementare joined by the dashed lines. Open and solid symbols represent odd and even elements from Z = 25 to Z = 36. The bold dashed curve joining the N = 50 proton-removed isotopes ( Se, As, Ge, Ga, Zn, and Cu are labeled with mass number) is obtained from a fit. The curveallows extrapolation of the production estimates of very neutron-rich nucleus such as Ni. N=42 N=43 N=44 N=45 N=46 N=47
30 351 30 351
N=48N=48
30 35
N=49
30 35
N=50
Nuclear charge Z ( EPAX ) s / B e K r + s FIG. 12: Ratios of Kr+ Be fragment experimental cross sections to EPAX [21] (open symbols)and to our modified EPAX Kr (solid symbols) predictions plotted as a function of nuclear charge, Z , for 42 ≤ N ≤
50 isotones.50 isotones.