Measurement of the equilibrium charge state distributions of Ni, Co, and Cu beams in Mo at 2 MeV/u: review and evaluation of the relevant semi-empirical models
P. Gastis, G. Perdikakis, D. Robertson, R. Almus, T. Anderson, W. Bauder, P. Collon, W. Lu, K. Ostdiek, M. Skulski
MMeasurement of the equilibrium charge state distributions of Ni, Co, and Cu beams in Moat 2 MeV / u: review and evaluation of the relevant semi-empirical models. P. Gastis a,b , G. Perdikakis a,b,c , D. Robertson b,d , R. Almus a , T. Anderson b,d , W. Bauder d , P. Collon b,d , W. Lu b,d , K. Ostdiek b,d , M.Skulski b,d a Department of Physics, Central Michigan University, Mt. Pleasant. MI 48859, USA b Joint Institute for Nuclear Astrophysics: CEE, Michigan State University, East Lansing, MI 48824, USA c National Superconducting Cyclotron Laboratory, Michigan State University, East Lansing, MI 48824, USA d Department of Physics, University of Notre Dame, Notre Dame, IN 46556, USA
Abstract
Equilibrium charge state distributions of stable Ni, Co, and Cu beams passing through a 1 µ m thick Mo foil were measuredat beam energies of 1.84 MeV / u, 2.09 MeV / u, and 2.11 MeV / u respectively. A 1-D position sensitive Parallel Grid AvalancheCounter detector (PGAC) was used at the exit of a spectrograph magnet, enabling us to measure the intensity of several chargestates simultaneously. The number of charge states measured for each beam constituted more than 99% of the total equilibriumcharge state distribution for that element. Currently, little experimental data exists for equilibrium charge state distributions forheavy ions with 19 (cid:46) Z p , Z t (cid:46)
54 (Z p and Z t , are the projectile’s and target’s atomic numbers respectively). Hence the success of thesemi-empirical models in predicting typical characteristics of equilibrium CSDs (mean charge states and distribution widths), hasnot been thoroughly tested at the energy region of interest. A number of semi-empirical models from the literature were evaluated inthis study, regarding their ability to reproduce the characteristics of the measured charge state distributions. The evaluated modelswere selected from the literature based on whether they are suitable for the given range of atomic numbers and on their frequentuse by the nuclear physics community. Finally, an attempt was made to combine model predictions for the mean charge state,the distribution width and the distribution shape, to come up with a more reliable model. We discuss this new ”combinatorial”prescription and compare its results with our experimental data and with calculations using the other semi-empirical models studiedin this work. Keywords: charge state distributions, semi-empirical models, heavy ion beams, molybdenum foil, gas cell windows
1. Introduction
The charge state distributions (CSDs) of heavy ions oftenhave to be considered in accelerator design and in the devel-opment of stable and radioactive isotope spectrometers. An im-portant need of information on charge state distributions ariseswith the use of recoil mass spectrometers (such as the electro-magnetic mass analyser, EMMA [1] at TRIUMF, Canada, andthe fragment mass analyser, FMA [2] at Argonne National Lab-oratory) to study nuclear reactions in inverse kinematics. Thesesystems use electromagnetic fields to filter out the reaction re-coils from the unreacted beam and rely critically on the fact thations with di ff erent m / Q ratios (where m is the mass of the ionand Q its charge) follow in principle di ff erent trajectories insidea field. In experiments with recoil spectrometers the usage ofgas-cell targets is common. As a beam of recoil products passesthrough the windows of a gas cell it interacts with electronsin the material via atomic charge-exchange processes (electroncaptures and losses). A CSD associated with the material andthickness of the window is observed for the particles exiting the ∗ Corresponding authors
Email addresses: [email protected] (P. Gastis), [email protected] (G. Perdikakis) target. The di ff erent charge states of the recoils follow di ff er-ent trajectories inside systems employing electric or magneticfields. Due to the typically limited acceptance of systems suchas magnetic dipoles, only a small number of charge-states (insome cases only one) can be transmitted through them and hitthe detector for a given field setting. Hence, to determine re-action yields using a recoil spectrometer, one needs to knowtheir charge state distribution. Molybdenum (Mo) foils are ex-tensively used as windows in gas-cells since they o ff er variousadvantages. The high resistance of Mo under mechanical stress(large Young’s modulus [3]) allows the achievement of high gaspressures using thin windows. Furthermore, Mo may reducesignificantly the background from fusion evaporation reactionscompared to a lower-Z foil.In the present study we measured the CSDs of stable Ni, Co, and Cu beams while passing through 1 µ m Mo foils(Z t = / u, 2.09 MeV / u,and 2.11 MeV / u for the Ni, Co, and Cu respectively. The re-sults of this study were used to check the agreement of semi-empirical models developed for heavy ions to the experimen-tal CSDs. In the following we present a brief description ofthe methods for the calculation of CSDs (subsection 1.1), theformalism for the semi-empirical models considered (subsec- Preprint submitted to NIM B September 26, 2018 a r X i v : . [ nu c l - e x ] M a r ion 1.2), the experimental procedure followed to extract CSDsat the University of Notre Dame (Section 2), our results in adetailed comparison and analysis of the successes and short-comings of various semi-empirical models (Section 3), and aconclusion and outlook (Section 4). The calculation of CSDs for systems with Z > / target combination of interest, a com-plete set of electron capture and loss cross sections must beknown [4]. Currently, a number of techniques for the calcula-tion of cross sections (e.g continuum distorted wave approxi-mation, plane-wave Bohr approximation) and the treatment ofthe electron exchange processes (e.g quasiground-state model,three charge-state model, etc.) are adopted by computer codesin order to numerically solve the problem. For example, thecodes CHARGE and GLOBAL have been designed for thecalculation of equilibrium and non-equilibrium CSDs, at ener-gies above 100 MeV / u, for heavy projectiles with atomic num-ber ( Z p ≥
30) in solid targets [5]. For intermediate energiesabove 10 MeV / u (and up to about 30 MeV / u), the programETACHA [6] calculates the evolution of charge state distribu-tions for ions with up to 28 electrons; the program has also beenchecked at energies around 2 MeV / u [7]. Finally, the code CS-Dsim has been developed at calculation of equilibrium and non-equilibrium CSDs for beams and recoils in gas targets [8]. Gen-erally speaking, for all the codes mentioned above the ability ofCSD prediction is limited; theoretical calculations on electronexchange cross sections don’t have the required accuracy, soexperimental values must be used instead. Calculated CSDsin regions (energies and atomic numbers) where cross sectionmeasurements have been done or where the adopted approxi-mations are correct (e.g at high energies) are more likely to beaccurate than outside of these regions.Apart from computational methods such as those describedabove, semi-empirical models are widely used to calculate equi-librium CSDs for practical applications (e.g on LISE ++ [9]).Important parameters of the distributions, such as the meancharge state and the distribution width, are given by empiri-cal formulas based on experimental data. Following that, sym-metric (Gaussian) or asymmetric functions are used to modelthe distribution and to calculate the equilibrium fractions. Themain advantage of semi-empirical models is the simplicity andspeed of the calculations for any system. However, the calcu-lations are strictly limited to equilibrium CSDs and for systems(ion / target) for which experimental data exist. In the energyregion up to 10 MeV / u for projectile and target atomic num-ber combinations 19 (cid:46) Z p , Z t (cid:46)
54 the semi-empirical modelsare expected to more accurately describe the charge state dis-tributions than the detailed codes described earlier (CHARGE,GLOBAL, ETACHA). However, their agreement with experi-ment has not been tested extensively since very few experimen-tal data exist for that region.
So far there is no quantitative theory to calculate from firstprinciples the mean charge state of heavy ions passing throughsolids. However, in the last decades, experimental data onequilibrium CSDs have been accumulated in databases, suchas [10], informing the development of semi-empirical calcula-tions. Di ffi culties are also encountered regarding the distribu-tion widths. For high-Z ions, the electron capture and loss crosssections change significantly across shell closures giving rise toobvious asymmetries in the CSD data (shell e ff ects) [11, 12].For such CSDs the notion of a single distribution width d isnot valid any more. However, since an analytical calculationof CSD for a heavy ion is very challenging, single widths arestill in use in order to approximately model charge distributions.It is common practice to calculate experimental widths assum-ing a Gaussian distribution (see Eq. 26 below), even for thosecases for which non Gaussian distributions are expected. Nev-ertheless, based on the existing experimental data, a number ofsuch empirical models have been developed for the estimationof equilibrium distribution widths in gaseous and solid targets.A number of semi-empirical models from the literature wereconsidered in this study, based on their validity in the relevantatomic number, energy range, and on their frequent use by thenuclear physics community to predict characteristics of CSD.In this section we will briefly present the basic formulation ofeach of these models and comment on their performance.The most recent semi-empirical model has been developedby G. Schiwietz et al. [13]. A many-parameter formula hasbeen fitted on data for about 840 experimental CSDs at vari-ous energies and atomic number ranges for the projectile andtarget. More specifically the data includes CSDs from solidtargets with 4 ≤ Z t ≤
83 (although more than 40% of the experi-mental distributions corresponds to C foils), and projectile ionswith 1 ≤ Z p ≤
92, at energies up to ∼
50 Mev / u. The formulaextracted by the fitting procedure is: q = Z p (8 . x + x )0 . / x + + . x + x (1)where: x = c ( ˜ υ/ c / . + . / Z p , (2) c = − . e − Z t / e − ( Zt − Zp )29 , (3) c = + .
03 ˜ υ ln ( Z t ) , (4)with the so-called scaled velocity ˜ υ , given in terms of the pro-jectile’s velocity υ p and Bohr velocity υ B , by:˜ υ = Z − . p υ p /υ B . (5)According to their study, deviations from experiment on themean charge-state values, q , are of the order of 2%. For thedistribution widths, G. Schiwietz et al. in [14] proposes a rela-tion of the form: d = w [ Z − . p Z . − . Z p t f ( q ) f ( Z p − q )] − (6)2here w is a scaled width and f ( x ) = (cid:113) ( x + . Z . p ) / x . (7)In [14], the width w is plotted versus the number of bound elec-trons, N b , of the projectile where N b = Z p - q exp . . According tothe scaled solid-state data, w ≈ b ≈ ff erent approach, (but also based on available experi-mental data) J.A Winger et al. [15] developed a phenomeno-logical parameterised formula for the mean charge state whichhas the form: q = Z p [1 − exp ( (cid:88) i = α i X i )] (8)where the reduced velocity X is given in terms of the beam’skinematic factor β , as: X = β/ . Z . p (9)The parameters α i are defined analytically in the way pre-sented in [15]. No additional information is given about themodel’s performance and agreement with experiment. In thesame study, a phenomenological parametrization for the widthsis also derived. According to this model: d = exp ( i = (cid:88) i = α i ( lnX ) i )[1 − exp ( i = (cid:88) i = β i ( lnZ t ) i )] (10)The parameters α i and β i are di ff erent in this formula from thosein the previous equation (for details see ref. [15]). In order toavoid large deviations due to shell e ff ects they used experimen-tal data at energies above ∼ / u. The energy of ions in thepresent study is outside this limit and hence, widths calculatedwith Winger’s formula are expected to show worse agreementwith experiment than some of the other more suitable modelswe considered.In an older study, Nikolaev and Dmitriev (ND) [16] deriveda semi-empirical formula for the mean charge states, accordingto which: q = Z p [1 + ( υ/ Z α p υ (cid:48) ) − / k ] − k (11)where α = k = υ is the projectile’s velocity, and υ ’ = cm / sec. Eq. (11) is designed to reproduce data withZ p (cid:38)
20, at energies of 5 to 200 MeV, mainly on C targets.Deviations from experiment on the q values do not exceed 5%,according to [16]. In the same study, an improved formula forthe distribution widths optimized for solid ion beam strippers isalso presented: d = d [ q [1 − ( q / Z p ) / k ]] / (12)where d = = p (cid:46)
37 at energies above 20 MeV, as discussed in [4].Based on the ND model, E. Baron et al. [17] developed theirown empirical formula to predict the average charge states q , for ion species in the range 18 ≤ Z p ≤
92 and energies up to 10.6MeV / u. According to their improved model: q = Z p [1 − Cexp ( − . β/ Z . p ] × [1 − exp ( − . + . Z p − . Z p )] . (13)where C = p > / u and C = + p ,for E p < / u. Their empirical formula for the widths hasthe form: d = (cid:113) q (0 . + . Y − . Y ) (14)where Y = q / Z p . The width model was designed to be more ac-curate for Z p >
54 at energies above 1.3 MeV / u.In addition to the above works for the mean charge state andwidth of the charge state distribution, in the present study wealso considered four more empirical models that provided for-mulas either for the mean charge state only, or for the distri-bution width. In particular these are: the work of H. D. Benz[4] for the distribution width, and the works of K. Shima et al.[18], To and Drouin [19], and A. Leon et al. [20] for the meancharge state. These models are presented in detail below; H.D. Benz, based on Nikolaev-Dmitriev’s work [16], provides thefollowing simple relation for the distribution width d : d = . Z / p (15)which provides a fair agreement with experimental data forheavy ions, up to Uranium, in Carbon and Formvar foils, atenergies below 80 MeV [4]. K. Shima et al. [18] developed amodel for the mean charge states for a large range of ion species(Z p ≥
8) in solid targets with 4 ≤ Z t ≤
79 at energies below 6MeV / u. According to his study: q = Z p [1 − exp ( − . X + . X − . X )] × [1 − . Z t − √ X + . Z t − X ] , (16)where the scaled velocity X is given by Eq. (9). The secondpart in Eq. 16 is actually a correction term for the non carbonsolid targets. This model reproduces the experimental data ofZ p ≥
14 with an agreement of ∆ q / Z p < q = Z p [1 − exp ( − υ/υ (cid:48) Z . p )] . (17)Even though the formulas in the work of To and Drouin werenot formulated to reproduce heavy ion / target combinations,their work was included in the current study since it is a deriva-tive of the generally successful Nikolaev and Dmitriev model.Finally, A. Leon et al. [20], based on Baron’s work [17], re-formulated the mean charge state expression by multiplyingEq. (13) with a suitable correction factor g’(Z t ,Z p ). This ledto improved fits of experimental data at energies of 18 MeV / u ≤ E p ≤
44 MeV / u for heavy ions with 36 ≤ Z p ≤
92 in varioussolid targets (4 ≤ Z t ≤ g (cid:48) ( Z t , Z p ) = [(0 . + . exp ( − . Z t )) ++ (0 . − . exp ( − . Z t )) v p Z . p ] (18)where υ p is the projectile’s velocity. To succesfully calculate charge state fractions, apart from themean charge state and the width, a model of the shape of thedistribution is also needed. At low and intermediate projectilevelocities in light gaseous and solid targets (Z t (cid:46) F q = ( d √ π ) − exp [ − ( q − q ) / d ] (19)where d is the distribution width, q is the charge of each state(an integer number), and ¯ q is the mean charge state of the distri-bution (in general a real number). For higher projectile veloc-ities in the same targets, typically for cases in which the meancharge state is very close to the Z p , the CSDs become asym-metrical (these asymmetries are explicitly dependent on the ve-locity and are not related with the shell e ff ects that we will dis-cuss next). For those distributions, Baudinet-Robinet et al. [21]proposed a distribution function extracted from a reduced χ distribution: F t = [2 ν/ Γ ( ν/ − t ν/ − e − t / (20)where the chi-squared variable is connected to the charge q,mean charge q , and width d as: t = c(Z p + = p + q ) / d and ν = c(Z p + q ).The CSDs of heavy projectiles, especially in heavy gasesand solids are asymmetric mainly due to atomic shell struc-ture. Those cases are far from Gaussian or χ distributions.K. Shima et al.[22], proposed the composite of two Gaussianfunctions with the same centroid but di ff erent standard devia-tions (widths) to emulate the asymetrical distributions of exper-iment. The now di ff erent left and right widths are associatedwith the di ff erent atomic shells. The usage of such a functionin the calculation of CSDs requires a model for the estimationof double widths which hasn’t been developed so far. Anotherapproach for treating the shell e ff ects has been proposed by R.O. Sayer [23] who introduced a modified Gaussian distribution: F q = F m exp [ − . t / (1 + (cid:15) t )] (21)where t = (q - q ) / ρ and F m is the fraction of the most intensecharge state q . The shell e ff ects can be satisfactorily repro-duced (even if they are not explicitly taken into account in Eq.21) if the proper values for ρ and (cid:15) are chosen. These values canbe extracted by fitting a suitable function of Z p and projectilevelocity, β c, on experimental data. Since there is not a satisfac-tory amount of data in the heavy ions region (Z p , Z t >
20) theagreement of this method to experimental CSDs may be some-what limited. Furthermore, the F m values must been known inadvance. Because of these limitations on Sayer’s and Shima’s Figure 1: Schematic overview of the FN Tandem acceleratorand the Accelarator Mass Spectroscopy beamline at the Nu-clear Science Laboratory of the University of Notre Dame. Themeasurements in this work made use of the MANTIS Browne-Buechner Spectrograph, the Multi-purpose Rotational Scatter-ing Chamber and the FN Tandem. The Wien filter shown in thisfigure upstream of the scattering chamber was not required andwas not used in the experiment.formulas, Eq. (19) and Eq. (20) are generally preferred by thecommunity for the calculation of CSDs (within their applica-tion limits) since they can be directly be combined with a largevariety of semi-empirical formulas for the mean charge state q and the distribution width d . In this work we followed thesame logic and decided to leave out the works of Sayer andShima [23],[22] from our comparison to experiment. Hence,for the modeling of the charge-state distributions we used onlythe Gaussian and the Baudinet-Robinet reduced χ distributionshapes described above.
2. Experimental Procedure and Data Analysis
The experiment took place at the Nuclear Science Labora-tory (NSL) of the University of Notre Dame. The incidention beams were accelerated by the 11MV FN Tandem Van deGraa ff accelerator. The accelerator mass spectrometry (AMS)beamline guided them into the Multi-purpose Rotational Scat-tering Chamber at the object of the MANTIS spectrograph (Fig.1). For our measurements the spectrograph was set at a 0 ◦ anglewith respect to the beam axis. Inside the scattering chamber aFaraday cup and three 1 µ m thick Mo foil targets, were mountedon a movable metallic frame with five target positions. The fifthposition was left blank so that the beam could pass through thechamber without interacting with the Mo foil. This setting wasused during beam tuning. After the target ladder, the beam ionscould enter the spectrograph where the di ff erent charge statesof the beam could be separated by the magnetic field.4igure 2: Trajectories of the di ff erent charge states of the sameelement in the spectrograph magnet. The PGAC detector is po-sition sensitive along the y-axis. The charge states are sepa-rated from each other by their charge-dependent position on thePGAC detector. The limit in the detector’s active region allowsthe measurement of only those charge states that have a suitabletrajectory radius in the magnet.A Parallel Grid Avalanche Counter (PGAC) detector, withactive region of 46 x 10 cm, was mounted on a set of rails onthe top of the spectrograph magnet. At this point the bent beamleaves the magnet vertically as it is shown in Fig. 2. By usingthe PGAC it was possible to measure a number of charge statessimultaneously since the detector is position sensitive. A moredetailed description of the detector’s operation can be found inprevious studies of D. Robertson et al. [24, 25].The full charge-state distribution for each beam would not fitinside the magnetic spectrograph’s acceptance in a single mag-netic field setting. Therefore, a gradual step-by-step increaseof the applied magnetic field was used to scan all detectablecharge states using the detector’s active region. In this process,by changing the field from lower to higher values the charge-states would be registered by the PGAC detector from higher tolower charge, since: | (cid:126) B | ∝ I magnet ∝ q (22)where (cid:126) B is the magnetic field, I magnet is the current supplied tothe spectrograph magnet, and a given trajectory radius throughthe magnet is assumed. Its important to mention that no fo-cusing elements were used before or after the dipole spectro-graph; so all charge states were transported to the detector withthe same focusing characteristics as the beam along the disper-sive direction of the magnet. The detector’s length allowed themeasurement of only 4 to 6 charge states at each step. As canbe seen in Fig. 3 the charge states appeared in the spectra asdi ff erent peaks along the horizontal axis which represents theY-position inside the PGAC. Scattering in the Mo foil is a con-tributing factor in the observed widths of the peaks.As mentioned before, because of the large number of chargestates in each distribution, the measurements were performed insteps. Between two consecutive steps some charge states were Figure 3: Charge-states of Cu taken with the Cu beam. Thespectrum corresponds to a single magnetic field setting. Thechannel number represents position along y-axis in the PGACdetector (0 cm to 46 cm). The intensity of each charge state isproportional to each peak’s area.chosen to be used as common references allowing the cross-normalization of the intensities of all charge states along thevarious spectra. To deduce the charge-state distribution fromthe data the following procedure was used: For each charge-state q we defined the relative fraction R q , calculated with re-spect to a reference state q re f . : R q = I q I q ref (23)where I q is the intensity of the state q and I q ref . is the intensity ofthe reference charge state. By using the common charge statesbetween adjacent steps as references we were able to calculate(for each distribution) all the relative fractions with respect toa single state. Having normalized the relative fractions in thisway, the net fractions F q could be extracted by: F q = R q (cid:80) q (cid:48) R q (cid:48) . (24)where the sum is over the total number of charge states in thedistribution. The uncertainties of the fractions were calculatedfrom the statistical errors in peak integration and taking intoaccount any overlap of adjacent peaks.Having deduced the fractions of all the charge states, themean charge of each distribution was calculated by: q = (cid:88) q qF q (25)and the distribution widths (assuming Gaussian distributions)by: d = [ (cid:88) q ( q − q ) F q ] / (26)5or each distribution measured in this experiment the regis-tered charge states did not constitute 100% of the whole CSD.The very low intensity charge states fell below the minimumdetection limit of the experimental set-up. Moving toward thehigher charge states the intensity drops very fast since the elec-tron loss cross section changes significantly; due to the finitedispersion of the magnet the high charge states (low m / Q ra-tio) are expected to be closer to each other, and so, significantlymore overlapped. These factors limited our ability to measurehigher charge states in a reasonable time. Regarding the lowercharge states, no reliable data could be taken for charge stateswith F q < − % due to the post-foil interactions of the beamions with the residual gas in the beamline vacuum. In such in-teractions it is expect that the electron capture cross sectionsare higher than the electron loss since the mean charge statesafter the Mo foil will tend to decrease in the residual air. Evenby assuming -according to our calculations for the experimen-tal beamline used- that only 0.1% of the beam ions will inter-act with the residual gas, the e ff ect on the intensity of the lowcharge states with F q < − % is still significant due to the elec-tron captures on higher charge states (especially on those withF q > ff ects are hard to be estimated without fur-ther measurements or knowledge of the relevant cross sections.The significance of the systematic uncertainty induced to thecharge state fractions due to the missed charge states was es-timated through a sensitivity test. Three more charge-stateswere added in each CSD and the e ff ects on the charge frac-tions were calculated. The additional charge states had frac-tions F q = − % in order to maximize the e ff ects. The chargestate fractions varied as a result of this procedure by a factorwhich was found to be at most 0.09%. Statistical errors comingfrom other contributing factors such as peak integration, fluctu-ated within a range of 1% to 41%. As a result, the error due tothe missed charge states was considered negligible.During the measurements no m / Q interferences from sec-ondary particles (reaction products) were present. All the nu-clear reaction channels from the interaction of the beams weused (Ni, Co, and Cu) with the Mo were found to have thresh-olds above 132 MeV while the beam energies we used wereup to 125 MeV. Furthermore, contaminations in the beam fromnuclear reactions with the carbon foil in the accelerator’s ter-minal were eliminated by the 90 degree beam analysis mag-net. Uncertainties due to possible pile-up were also negligible.Throughout the measurements the count rate in the detector wasmonitored and was found to be in no case more than of the orderof 5000 counts / sec. These rates are too small to induce pile-upsince the signal processing time in the counter detector used isof the order of a few microseconds for each particle.
3. Results and Discussion
In Fig. 4 the CSDs measured in the current study are pre-sented. Furthermore, Table 1 includes the fractions of all themeasured charge states in detail. The statistical error on thecharge state fractions fluctuated between 1% and 6% in mostof the charge states. The largest uncertainties were observedfor the 22 + ( ∼ + ( ∼ + ( ∼ ff ects. This asymmetry is evident by examining the ra-tios of the fractions for each CSD. In the case of a symmet-ric (Gaussian-like) equilibrium CSD, the logarithm of the ra-tios F q + / F q is linearly varying with the charge q. This is be-cause, at equilibrium, the ratios σ q , q + / σ q + , q (where σ q , q + isthe electron loss cross-section at the charge-state q and σ q + , q is the electron capture cross-section at q +
1) are approximatelyproportional to e − q and F q + / F q = σ q , q + / σ q + , q [27]. The abovestatement implies that the single electron exchanges are domi-nant. However, when shell transitions or multiple-electron ex-changes occur the relationship between ln(F q + / F q ) and q is notlinear, resulting in asymmetrical CSDs. In Fig. 5, the discon-tinuities at q = + , q = + , and q = + (in Co, Ni, and Cu re-spectively) are consistent with the L-M shell transitions as canbe shown by examining the corresponding electronic configu-rations [28]. The fractions of charge states that correspond toclosed shells, or sub-shells, are significantly enhanced resultingin ”kinks” on the plot. For the case of Co, the discontinuity atq = + seems to correspond to the 2 s -2 p sub-shell transition.In Table 2, all the calculations of the mean charge q usingvarious semi-empirical models are presented. The experimen-tal values have been calculated by Eq. (4). It is important tonote that the mean charge of the distribution doesn’t have tobe an integer number. The maximum charge state (the mostintense) is the integer which is closer to the mean value. Interms of their predictive power for the maximum charge state,Schiwietz et al. and Shima et al. o ff er the best agreement toexperimental data. The agreement of the other models exam-ined in this work is within ± q were used in all the calculations.By neglecting shell e ff ects, most of the models diverged from6igure 4: Charge state distributions of Ni (black inverted tri-angles and line), Co (red circles and line), and Cu (blue up-right triangle and line) beams in a 1 µ m thick Mo foil. No errorbars are visible for statistical errors less than 3% due to the sizeof the point markers. The experimentally deduced mean chargestates for each distribution are q = q = q = + ,18 + , and 19 + for Ni, Co, and Cu respectively.Figure 5: Logarithm of the ratio of two adjacent charge statefractions as function of the charge of ions detected with theMANTIS spectrograph in the present study. Lines are used toguide the eye. Black inverted traingles and a black line corre-spond to Ni ions on Mo using the Ni beam, red circles and linecorrespond to Co ions on Mo using the Co beam, and blue up-right triangles and line correspond to Cu ions on Mo using the Co beam. Divergence from a straight line is attributed to shelle ff ects and multiple-electron exchange processes. All observeddistributions demonstrated such deviations which correspond tonon-symmetrical CSDs. Figure 6: Equilibrium CSD of Cu in Mo, as measured in thepresent study (blue line, upright triangles) and as reported by KShima et al [29] (magenta line inverted triangles). The distribu-tions are assigned to emerging projectile energies (after energyloss in the Mo foil). Mostly excellent agreement between the 2measurements is observed.the experimental widths. A systematic overestimation was ob-served on the values calculated with Nikolaev-Dmitriev (ND)and Betz models, while the Baron et al. and Winger et al. mod-els systematically underestimated the distribution widths. Themodel by Schiwietz et al. was shown to be in better agree-ment with experiment in reproducing the widths of the mea-sured CSDs with very small deviations. This success we at-tribute to the fact that this model was based on fitted data thatincluded a significant number of asymmetrical CSDs.The only available data in literature, relevant to the currentwork, are coming from a previous study by K. Shima et al. [29].Shima et al. used a Cu beam impinging on a Mo foil. The re-sults are plotted in Fig. 6 in comparison to data from the presentwork. Comparing the two data sets we see a mostly excellentagreement to each other considering the small energy di ff erenceof the two cases. The fractions of the charge states 19 + and 20 + are identical within the statistical errors while deviations are ob-served on charge-states with fractions lower than 15%. In termsof the mean charge state and the distribution width (see Tables2 and 3), the deviations are within the statistical errors. The results of this study support a systematically betteragreement with experimental data of the formulation of Schi-wietz et al. [14] for the distribution widths and of Winger etal. [15] for the mean charge states q . Combining these modelswith a Gaussian or a reduced χ distribution function, we ex-plored a more realistic way to reproduce the shape of CSDs inthe region of our experimental study (Z p ∼
28 and Z p = d and the7ean charge state q . Each of these calculations was performedin two variants; one assuming a Gaussian shaped charge statedistribution and one assuming a Baudinet-Robinet type reduced χ distribution. The results are presented in Fig. 7 (a-f) whileon Table 4 the ˜ χ values of the calculated CSDs are presentedas extracted from chi-square goodness of fit tests.From the comparison in Fig. 7 it is suggested that the com-binatorial model combined with a reduced χ distribution func-tion produces a qualitatively better description of the experi-mental data especially in the regions of low intensity chargestates near and at the tails of each distribution. At the higher in-tensity charge states around the mean no significant di ff erencebetween the reduce χ and Gaussian-shape charge distributionsis observed. Nevertheless, the overall agreement o ff ered by thecombinatorial model (combined with any of the two distribu-tion functions we discussed) is improved in comparison to theother models as demonstrated by the lower on average value of˜ χ (see last row of table 4).
4. Conclusion
Equilibrium charge state distributions of Co, Ni, and Cu beams passing through a 1 µ m thick Mo foil have beenmeasured. A variety of semi-empirical models for the meancharge state q and the distribution width d of equilibrium chargestate distributions were compared with our experimentally de-termined charge state distributions. Furthermore, our study sug-gests that an improved agreement of the calculated equilibriumCSDs to the experimental data in the region of study (Z p ∼ p =
42, and E (cid:118) / u) can be obtained by using a com-bination of models to describe the equilibrium CSDs. In this”combinatorial” description, the formulation of Winger et al.[15] is used to calculate the mean charge state q, the work ofSchiwietz et al. [14] is followed in the calculation of the dis-tribution width d , and a reduced χ or a Gaussian function canbe used to describe the shape of the charge state distribution.Despite the improved agreement of this ”combinatorial” modelit is still a phenomenological prescription with obvious limita-tions. Realistic first-principle based simulations of equilibriumand non-equilibrium CSDs of heavy ions would be the idealway to go forward. Until this is reliably possible, it would bebeneficial to extend the current set of experimental data to coverthe 19 (cid:46) Z p , Z t (cid:46)
54 region at various energies below 10 MeV / u. Acknowledgements
The authors would like to thank Dr. Oleg Tarasov for shar-ing details about the code LISE ++ [9] and for valuable discus-sions. The authors also acknowledge support from College ofScience and Technology at Central Michigan University. Thisresearch was supported by the National Science Foundation(PHY-1419765), Michigan State University and the Facility forRare Isotope Beams. ReferencesReferences [1] Barry Davids and Cary N. Davids. Emma: A recoil mass spectrometer forisac-ii at triumf.
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Atomic Data Nuclear Data Tables ,34:357–391, 1986. a) (b)(c) (d)(e) (f) Figure 7: Calculated CSDs using di ff erent semi-empirical models including our “combinatorial” formulation in comparison withexperimental data. The left column of the figure shows calculations assuming a reduced χ type shape of the distribution, whilethe right column shows calculations using a Gaussian shape. Figures (a,b) : Comparison of experimental data from this work tocalculations for a Co beam. Figures (c,d) : the same for Ni. Figures (e,f) : The same for Cu. In all figures, experimental dataare represented by black stars and line, the calculations using the work of Schiwietz et al (S in figure) are presented with red circlesand line, the calculations using the work of Winger et al. (W in figure) are presented by magenta inverted triangles and line, andthe combinatorial formulation introduced in this work (W + S) is presented with blue upright triangles and line. Values of χ for thecomparison are presented in table 4. 9able 1: Fractions percent (%) of all the measured charge states in the present study. Each row of the table corresponds to adi ff erent beam in Mo. Ion E(MeV / u) F F F F F F F F F F F Co 1.769 ± ± ± ± ± ± ± ± ± ± ± Ni 1.510 ± ± ± ± ± ± ± ± ± ± Cu 1.783 ± ± ± ± ± ± ± ± ± ± ± Table 2: Comparison of experimentally and theoretically determined values of the mean charge state q in Mo. Predictions of thevarious semi-empirical models considered in this work are shown. Each row of the table corresponds to a di ff erent beam. The lastrow gives for reference the experimental value from literature for Cu which is in agreement with our own measurement. Ion E (MeV / u) Exper. Schiwietz [13] Shima [22] Baron [17] Nik.-Dmit. [16] Drouin [19] Winger [15] Leon [20] Co 1.769 ± ± Ni 1.510 ± ± Cu 1.783 ± ± Cu 1.767 As reported by K. Shima et al. [29]
Table 3: Comparison of experimentally and theoretically determined values of the distribution width d in Mo. Predictions of thevarious semi-empirical models considered in this work are shown. Each row of the table corresponds to a di ff erent beam. The lastrow gives for reference the experimental value from literature for Cu which is in agreement with our own measurement. In order toproperly compare the predictions of the models with the experimental distribution width, the experimental value of q was used ascommon input in all calculations. Ion Target Experimental Schiwietz [14] Baron [17] Nik.-Dmit. [16] Winger [15] Betz [4] Co Mo (foil) 1.31 ± Ni Mo (foil) 1.33 ± Cu Mo (foil) 1.35 ± Cu Mo (foil) As reported by K. Shima et al. [29]
Table 4: ˜ χ values of the calculated CSDs. In the calculations were used either combination of models or single models for the q and d . For explanation of labels see text and figure 7: (W) : Winger et al., (W + S) : this work, (S) : Schiwietz et al., (G) : Gaussiandistribution, ( χ ) : Reduced χ distribution. W + S (G) W (G) S (G) W + S ( χ ) W ( χ ) S ( χ ) Co 2.48 7.22 2.58 3.85 11.31 6.17 Ni 1.74 7.20 17.85 1.81 5.92 22.08 Cu 5.33 16.45 4.12 4.66 9.85 4.93Average ˜ χ : 3.18 10.29 8.18 3.44 9.02 11.06: 3.18 10.29 8.18 3.44 9.02 11.06