Determination of angular distributions from the high efficiency solenoidal separator SOLITAIRE
L. T. Bezzina, E. C. Simpson, D. J. Hinde, M. Dasgupta, I. P. Carter, D. C. Rafferty
DDetermination of angular distributions from the high efficiencysolenoidal separator SOLITAIRE
L. T. Bezzina ∗ , E. C. Simpson, D. J. Hinde, M. Dasgupta, I. P. Carter and D. C. Rafferty Department of Nuclear Physics, Research School of Physics, Australian National University, Canberra 2601, Australia
A R T I C L E I N F O
Keywords :Nuclear fusionGas filled separatorEvaporation residuesSuperconducting solenoid
A B S T R A C T
A novel fusion product separator, based on a gas-filled 8 T superconducting solenoid has been de-veloped at the Australian National University. Though the transmission efficiency of the solenoid isvery high, precision cross section measurements require knowledge of the angular distribution of theevaporation residues.A method has been developed to deduce the angular distribution of the evaporation residues from thelaboratory-frame velocity distribution of the evaporation residues measured at the exit of the separator.The features of this method are presented, focusing on the example of S+ Y which is comparedto an independent measurement of the angular distribution. The establishment of this method nowallows the novel solenoidal separator to be used to obtain reliable, precision fusion cross-sections.
1. Introduction
Precise measurements of fusion cross sections are keyto advancing our understanding of heavy-ion fusion [1, 2].They have revealed the role of nuclear structure in enhanc-ing sub-barrier fusion cross sections [3, 4] and, via mea-surements of barrier distributions [5], demonstrated the im-portance of deformations [6] and surface vibrations [7]. Asthe field seeks to understand the observed above-barrier sup-pression of fusion [8], hindrance in deep sub-barrier fusion[2], and efforts continue to synthesise new super-heavy ele-ments to map out the so-called “island of stability”, precisefusion cross sections will continue to be critical.Experimental determination of fusion cross sections in-volves measurement of the product nuclei generated follow-ing the formation of a fused intermediate, the compound nu-cleus (CN). Two main decays modes are observed: fissioninto two fragments, or emission of nucleons (or clusters ofnucleons) to form an evaporation residue (ER). The balancebetween these decay modes depends on the properties of thecompound nucleus formed, namely, mass, charge, excitationenergy and angular momentum.The measurement of ER cross sections is particularlychallenging. Momentum conservation leads to an ER an-gular distribution that is strongly forward-focused, but thecross section is, at best, times smaller than elastic scat-tering at the same angles. The major challenge is then theefficient and effective separation of the evaporation residuesfrom the elastically scattered beam particles. For barrier dis-tribution measurements, which require cross sections withuncertainties of 1% or better, it is vital that the transmis-sion efficiency of the separator be known very well. At theDepartment of Nuclear Physics at the Australian NationalUniversity (ANU), separation is achieved using a separationsystem based around a gas-filled superconducting solenoid,called SOLITAIRE [9]. In this paper, we describe a methodfor deducing the angular distribution of the ERs from themeasured velocity distribution, and use this information to ∗ Corresponding author. Email: [email protected] deduce the transport efficiency of the device and thus deter-mine precise ER cross sections.This method was initially presented at the conferenceFusion17 and published in the proceedings of that confer-ence [10]. While the method remains similar in form, thispaper presents it with greater detail and tests it with greaterrigour, instilling greater confidence in the use of this methodto determine precise ER cross sections.
2. SOLITAIRE
The superconducting solenoid for in-beam transportand identification of recoiling evaporation-residues (SOLI-TAIRE) separates evaporation residues from the intenseflux of elastically scattered beam particles using the mag-netic field of the solenoid. The principles of operationand first measurements of SOLITAIRE are detailed byRodríguez [9], and as such, only a brief summary follows.The 14UD tandem accelerator at the ANU Heavy IonAccelerator Facility provides pulsed beams of the desiredenergy, which are transported to the SOLITAIRE targetchamber (as shown in Figure 1). The target chambercontains four silicon monitor detectors placed at ◦ , whichare used for normalisation purposes. Immediately prior tothe target chamber the beam passes through a thin carbonfoil, which separates the helium gas-filled region from theupstream beam line. The energy losses of the primarybeam in both the carbon foil and in the helium gas priorto the reaction target can be significant, particularly atnear-barrier energies, and must be taken into account. Thisis achieved by scattering the beam from a thin gold foilin the target position and measuring the change in energyin the monitor detectors with and without the helium gasand carbon foil present. For the benchmarking reactionsdiscussed in this paper the energy loss quantified in this waywas .
92 ± 0 . MeV.Following the reaction target, an on-axis Faraday Cupblocks the unscattered beam and any particles with labora-tory scattering angles 𝜃 𝑙 < . ◦ . The iron nose cone of the L. T. Bezzina et al.:
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Page 1 of 8 a r X i v : . [ phy s i c s . i n s - d e t ] J a n eam TargetMonitor Detector Blocking Discs and RodElasticallyScattered BeamEvaporationResidues MWPCsGas-FilledVolumeIron ConeFaraday CupCarbon EntranceFoil Figure 1:
Schematic diagram of SOLITAIRE. This figure is adapted from [9], and shows key components of the device, as wellas indicative radial trajectories of evaporation residues (purple dashed line) and elastically scattered beam particles (orange line).The shaded grey area indicates the iron shielding of the solenoid, including a ‘nose’ cone at the solenoid entrance. Also indicatedare two of the four monitor detectors used for normalisation to elastic scattering, and the two multiwire proportional counters(MWPCs) used for direct detection of evaporation residues. solenoid blocks all particles with 𝜃 𝑙 > ◦ . ERs and elasti-cally scattered beam ions within these angular limits proceedinto the solenoid.The solenoid acts as a thin converging lens for thecharged particles, separating them based on their magneticrigidity [9, 11]. Upon entering the solenoid the ERs andelastically scattered beam have the same average momentumand often have overlapping charge state distributions, andtherefore similar rigidity. For this reason, the solenoid boreis filled with a low pressure (usually ∼ ⟨ 𝑞 ⟩ , isdependent on the velocity and atomic number of the nuclei[12]. The significantly higher velocity of the elasticallyscattered beam nuclei bring them to a higher charge statethan the lower velocity ERs.With a higher charge state, the elastically scattered par-ticles have a lower rigidity and shorter focal length, and areintercepted by the blocking rod and discs placed along thesolenoid axis in the solenoid bore. The lower charge-stateERs exit the solenoid, and are brought to a focus furtherdownstream and are detected in SOLITAIRE’s two multi-wire proportional counters (MWPCs). The MWPCs recordposition, energy loss, and timing information of each event.Owing to the large solid angle of acceptance of thesolenoid (86 msr [9]) the transmission efficiency of thedevice is very high. This means SOLITAIRE has thecapability to measure ER cross sections with very highprecision. In order to accurately extract ER cross sections,however, a number of quantities need to be precisely known:1. the angular distribution of ERs leaving the target,2. the fraction of ERs entering the solenoid, 3. the fraction of ERs which are transported through tothe detectors,4. the fraction of ERs incident on the detectors which arerecorded in the data acquisition system, and5. beam normalisation through Rutherford scattering atthe monitor detectors.Throughout this paper, points 2. and 3. are combined todefine a ‘transport efficiency’, that is, the ratio of ERs exitingthe solenoid relative to those produced at the target. The pri-mary factor affecting the transport efficiency is the angularacceptance, defined by the Faraday Cup and iron nose cone.The probability that any given event with laboratory angle 𝜃 𝑙 is transported through the solenoid can be simulated us-ing Monte Carlo techniques, but the total transport efficiencyrequires knowledge of the angular distribution of ERs (point1). This angular distribution must be deduced from the ex-perimental measurements, as, while statistical model calcu-lations using packages such as PACE4 [13] are possible inprinciple, the input parameters of these calculations are notconstrained enough to precisely predict experimental angu-lar distributions.The complication in the present work is that any mea-surement of the ERs takes place after the solenoid, and istherefore already moderated by the transport efficiency. Thispaper specifically addresses this conundrum, and presents amethod to deduce both the transport efficiency of the deviceand the angular distribution of ERs, as described in the nextsection.
3. Method
We now discuss the method for extracting the transportefficiency of the solenoid and ER angular distribution from
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Page 2 of 8 he target. As the MWPCs are position sensitive, one mighthope to use the position information to directly reconstructthe initial angular distribution of the evaporation residues.However, this approach is sensitive to the details of the scat-tering and charge-changing interactions with the helium fillgas, which spread the trajectories of the transmitted ions.This also requires precise knowledge of the focal point ofthe ERs [14]. A new and more reliable approach has beenconcieved, to use the velocity distribution of the ERs. Thevelocity itself can be measured using the timing of the pulsedbeam and the MWPC signals. The velocity is unaffected onaverage by the interactions with the gas, and is therefore rep-resentative of the velocity distribution of the ERs at the reac-tion target, albeit filtered by transport through the solenoid.The laboratory velocity and angular distributions ofERs are linked via the distribution of events in the centre-of-momentum frame, demonstrated by the velocity vectordiagram shown in Figure 2. This diagram illustrates the netrecoil of an evaporation residue following the emission ofnucleons or nucleon clusters from the compound nucleus.The final ER then has some velocity and angle relative tothe beam axis in the centre-of-momentum frame, labelled 𝑣 𝑐 and 𝜃 𝑐 , respectively. Summed with the velocity of thecompound nucleus in the laboratory frame, 𝑣 𝑐𝑛 , the ER hasresulting laboratory velocity and angle ( 𝑣 𝑙 , 𝜃 𝑙 ), which areexperimentally measurable quantities.If one assumes some form for the probability of ER re-coils in the centre-of-momentum frame, that is, the distribu-tion of events with ( 𝑣 𝑐 , 𝜃 𝑐 ) any cross section (e.g., 𝑑𝜎 ∕ 𝑑𝜃 𝑙 , 𝑑𝜎 ∕ 𝑑𝑣 𝑙 , 𝑑𝜎 ∕ 𝑑 Ω ) can be reconstructed. Here, we write thisin terms of the unit normalised recoil velocity probabilitydistribution 𝑃 ( 𝑣 𝑐 ) , and its angular distribution 𝑊 ( 𝜃 𝑐 ) . Todemonstrate this relationship between 𝑃 ( 𝑣 𝑐 ) and 𝜎 𝐸𝑅 ( 𝑣 𝑙 ) wepresent the double differential cross section [15]: 𝑑 𝜎 𝐸𝑅 𝑑𝑣 𝑙 𝑑𝑣 𝑐 = 2 𝜋𝜎 𝐸𝑅 [ 𝑣 𝑙 𝑣 𝑐 𝑣 𝑐𝑛 ] 𝑊 ( 𝜃 𝑐 ) 𝑃 ( 𝑣 𝑐 ) , (1)where 𝜎 𝐸𝑅 is the total evaporation residue cross section andall other variables are as defined above and in Figure 2. Forthe rest of this paper, we assume the net emission (includ-ing all 𝑥𝑛 , 𝑝𝑥𝑛 and 𝛼𝑥𝑛 channels) is isotropic, as tests of thesensitivity of our results to an expected range of anisotropiesin 𝑊 ( 𝜃 𝑐 ) (using an upper limit determined from [16]) hadnegligible effects on the shape and normalisation of the finaldistributions. The procedure used to extract 𝜎 𝐸𝑅 and 𝑃 ( 𝑣 𝑐 ) from the velocity measurements made at the MWPCs is de-scribed below. The first component required is an estimation of thetransport efficiency of the solenoid for each ER with a givenlaboratory velocity ( 𝑣 𝑙 ) and angle ( 𝜃 𝑙 ). Note that there is adistinction between the transport efficiency for a particular 𝜃 𝑙 and 𝑣 𝑙 , denoted 𝜀 ( 𝜃 𝑙 , 𝑣 𝑙 ) , and the total transport efficiency 𝜀 𝑇 , which is integrated over all velocities and angles which θ l θ c v cn v c v l v cn = CN velocity θ l = ER angle v l = ER velocityIn the laboratory frame: θ c = ER angle v c = ER velocityIn the centre-of-momentum frame: Figure 2:
Velocity vector diagram showing the relationshipbetween velocities of compound nucleus (CN) and evaporationresidue (ER) in centre-of-momentum and laboratory frames.The compound nucleus velocity in the laboratory frame is 𝑣 𝑐𝑛 ,which is related to the ER velocity and angle in the laboratoryframe ( 𝑣 𝑙 and 𝜃 𝑙 , respectively) via the ER velocity and angle inthe centre-of-momentum frame of the compound nucleus ( 𝑣 𝑐 and 𝜃 𝑐 ). are emitted from the target. The calculations for 𝜀 ( 𝜃 𝑙 , 𝑣 𝑙 ) were performed with the Monte Carlo program Solix [17],developed at the ANU. The program simulates the passageof particles of specified mass, initial energy and anglethrough the solenoid. As well as accounting for the physicalobstructions of the blocking discs and rod (among others),it models the charge-changing collisions with the specifiedfill gas, and can count the number of particles incident onany specified area along the solenoid’s axis.The ERs were simulated as originating from a beam witha Gaussian profile with a width of 𝜎 = 0 . mm in both 𝑥 and 𝑦 (perpendicular to the 𝑧 of the beam axis) though it wasfound that the resulting efficiency map was largely insensi-tive to the precise size of the simulated beam spot. The rangeof velocities was chosen to correspond to the measured ve-locity distribution range, while the range of angles slightlyexceeded the angular acceptance of the iron nose cone of thesolenoid. The mean charge state of the ERs was assumedto depend linearly on the velocity [12]. While more com-plex models of the charge state’s velocity dependence ex-ist [12, 18], the average charge state in all of these models isvery nearly linear in velocity, as noted in Refs. [18, 19].The simulated transport efficiency for the ERs from the S+ Y reaction is presented in Figure 3, with the centralvelocity corresponding to that appropriate for 𝐸 𝑙𝑎𝑏 = 124 MeV. Due to the assumption that the mean charge state isproportional to the velocity, the ion rigidity and thus effi-ciency are independent of velocity. In angle, the Faradaycup stops all ions (direct beam and ERs) with 𝜃 𝑙 < . ◦ .The cut at 𝜃 𝑙 > ◦ is due to the angular acceptance of theiron cone of the solenoid. 𝑃 ( 𝑣 𝑐 ) With the transport efficiency 𝜀 ( 𝜃 𝑙 , 𝑣 𝑙 ) simulated, we thenneed to deduce the form of the ER recoil velocity distribu-tion in the compound nucleus frame, 𝑃 ( 𝑣 𝑐 ) . As outlinedby Weisskopf [20], the distribution of ER velocities in thecompound-nucleus rest-frame is expected to take the shape L. T. Bezzina et al.:
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Page 3 of 8 .6 0.65 0.7 0.75 0.8 0.85 0.9 ( deg r ee s ) l θ Labo r a t o r y ang l e , Velocity, v l (cm/ns) R e l a t i v e e ffi c i en cy l θ Laboratory angle, 00.20.40.60.8 R e l a t i v e e ff i c i c i en cy Figure 3:
The two-dimensional efficiency map (top), 𝜀 ( 𝜃 𝑙 , 𝑣 𝑙 ) ,of SOLITAIRE for the S+ Y reaction, is uniform in velocity 𝑣 𝑙 , but has a strong dependence on the laboratory angle, 𝜃 𝑙 .The bottom figure shows the efficiency as a function of lab-oratory angle only, in order to more clearly demonstrate theangular dependence of the efficiency. of one or more Maxwellian distributions. This expectationarises from a statistical description of the compound nucleus(CN) due to the high level-density of the accessible excitedstates. Different modes of evaporation (e.g. 𝑛 , 𝑝 , 𝛼 ) have adiffering energy threshold for emission, which is dependenton the Coulomb barrier between the emitted particle and re-maining residue. For each evaporation residue, a series ofemissions is likely to have occurred.These factors lead us to expect 𝑃 ( 𝑣 𝑐 ) to be composedof several Maxwellian functions, one with no offset from 𝑣 𝑐 = 0 , representative of 𝑥𝑛 emission, where there is nobarrier to emission. There could also be Maxwellians rep-resenting 𝑝𝑥𝑛 , 𝛼𝑥𝑛 , 𝛼𝑝𝑥𝑛 and so on, each of which wouldhave been offset from zero proportional to the magnitude ofthe Coulomb barrier faced by the charged particle/s emitted.In our case, the compound-nucleus rest frame distribu-tion is parameterized using the sum of two Maxwellians,the first representing neutron and proton emission, while thesecond simulated 𝛼𝑥𝑛 -emission (and therefore being offsetfrom zero). The initial parameters were estimated using a abootstrap extraction of 𝑃 ( 𝑣 𝑐 ) from 𝜎 𝐸𝑅 ( 𝑒𝑥𝑝 )( 𝑣 𝑙 ) , following amethod outlined in [14], based on Equation 1. More com-plex forms for 𝑃 ( 𝑣 𝑐 ) are discussed in Section 4.2. Centre-of-momentum(parameterised)distribution Laboratory(measured)distributionTransporte ffi ciency ε ( v l , θ l ) σ ER ( v l , θ l ) σ ER ( ε ) ( v l ) σ ER ( exp ) ( v l ) σ ER ( θ l ) χ σ ER ( v c ) Figure 4:
Flowchart of the method for extracting precise an-gular distributions from measurements made with the SOLI-TAIRE device, with a minimised schematic of the device in-cluded for context. The flowchart shows the routine describedin Section 3, including the initialisation step for estimating ini-tial parameters of the 𝜎 𝐸𝑅 ( 𝑣 𝑐 ) distribution, and the final trans-formation to the angular distribution 𝜎 𝐸𝑅 ( 𝜃 𝑙 ) . Having chosen a parameterised form of 𝑃 ( 𝑣 𝑐 ) , we thenuse a correction procedure to deduce the total transport ef-ficiency of the solenoid 𝜀 𝑇 and therefore the evaporationresidue angular distribution. First, the timing information(corrected for energy loss in the He fill gas) from the frontMWPC is used to produce a distribution of events in labo-ratory velocity 𝑣 𝑙 . This experimental distribution, measuredat the MWPCs, is filtered by the SOLITAIRE transport ef-ficiency. Next, taking the assumed form for 𝑃 ( 𝑣 𝑐 ) , a the-oretical double differential cross section 𝑑 𝜎 𝐸𝑅 ∕ 𝑑𝜃 𝑙 𝑑𝑣 𝑙 isgenerated using: 𝑑 𝜎 𝐸𝑅 𝑑𝜃 𝑙 𝑑𝑣 𝑙 = 2 𝜋𝜎 𝐸𝑅 sin 𝜃 𝑙 𝑊 ( 𝜃 𝑐 ) 𝑃 ( 𝑣 𝑐 ) 𝑣 𝑙 𝑣 𝑐 . (2)This represents the distribution of ERs emitted fromthe reaction target, which then needs to be filtered bythe transport efficiency in order to compare to the mea-sured laboratory velocity distribution. The cross section 𝑑 𝜎 𝐸𝑅 ∕ 𝑑𝜃 𝑙 𝑑𝑣 𝑙 is simply multiplied by the transport effi-ciency 𝜀 ( 𝜃 𝑙 , 𝑣 𝑙 ) bin-by-bin. Integrating the result over thelaboratory angle 𝜃 𝑙 then gives an efficiency-filtered velocitydistribution, 𝜎 𝐸𝑅 ( 𝜀 ) ( 𝑣 𝑙 ) , which allows a direct comparisonwith the laboratory velocity distribution measured at theexit of the solenoid, 𝜎 𝐸𝑅 ( 𝑒𝑥𝑝 ) ( 𝑣 𝑙 ) , and evaluation of the 𝜒 of these two distributions. This series of transformations,beginning from the creation of the double differential crosssection 𝑑 𝜎 𝐸𝑅 ∕ 𝑑𝜃 𝑙 𝑑𝑣 𝑙 , is then optimised by iterating overthe parameters of 𝑃 ( 𝑣 𝑐 ) in order to minimise the 𝜒 . Theprocedure is illustrated in Figure 4. Examples of the final,optimised velocity distribution for 𝐸 𝑏𝑒𝑎𝑚 = 112 and 124MeV are shown alongside the experimentally measured ve-locity distributions in Figure 5. In both cases the agreementis very good. L. T. Bezzina et al.:
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Page 4 of 8 .55 0.6 0.65 0.7 0.75 0.8 0.85(cm/ns) l Velocity, v − −
10 110 ( m b / b i n ) l /d v E R d σ Experimental DataFit Result E beam = 112 MeV ( m b / b i n ) l /d v E R σ d − −
10 110
Experimental DataFit Result
Velocity, v l (cm/ns) E beam = 124 MeV Figure 5:
A comparison of the laboratory-frame velocity dis-tributions at the beam energies 𝐸 𝑏𝑒𝑎𝑚 = 112 MeV (top)and 𝐸 𝑏𝑒𝑎𝑚 = 124 MeV (bottom). The experimental data, 𝜎 𝐸𝑅 ( 𝑒𝑥𝑝 ) ( 𝑣 𝑙 ) , are shown as red points, while the result of theminimisation routine, 𝜎 𝐸𝑅 ( 𝜀 ) ( 𝑣 𝑙 ) , is shown as a black histogram.The experimental distribution is naturally filtered by the SOLI-TAIRE transport efficiency, while the result of the minimisationroutine is filtered by the simulated transport efficiency.
4. Results 𝑃 ( 𝑣 𝑐 ) The first confirmation of the suitability of our procedurecomes from an examination of the final, optimised 𝑃 ( 𝑣 𝑐 ) .While the offset of the first Maxwellian was fixed at 0 cm/ns(expected to represent 𝑥𝑛 and 𝑝𝑥𝑛 emission), no other pa-rameters were fixed or limited. The expected value of theoffset of the second Maxwellian physically corresponds tothe minimum recoil velocity an ER can be expected to have,assuming a single alpha is emitted. In the scenario with thelowest centre-of-momentum energy, this corresponds to thebarrier energy between the alpha particle and resultant ER,which is then shared between the two nuclei in proportionto their masses. Using a São Paulo potential for the nuclearpart of the barrier potential, the expected offset for 𝛼 emis-sion from the CN was then calculated to be 0.0642 cm/ns.The iterative correction routine finds an optimised offset of .
073 ± 0 . cm/ns, which is in reasonable agreement withthe calculated value. The final, optimised 𝑃 ( 𝑣 𝑐 ) distributionis shown in Figure 6. c Velocity, v ) c P ( v Figure 6:
The optimised centre-of-momentum frame velocitydistribution for the case of 𝐸 𝑏𝑒𝑎𝑚 = 124 MeV. The solid lineshows the total ER distribution, including all emission modes.The dashed lines show the two component ER distributions,one peaked at 𝑣 𝑐 ∼ 0 . cm/ns representing ERs from 𝑥𝑛 and 𝑝𝑥𝑛 emission, while the other represents ERs resulting from 𝛼𝑥𝑛 emission. We next compare results for the extracted angular dis-tribution to the independent velocity filter measurements ofRef. [21]. Due to energy losses in the carbon foil and heliumgas as discussed in Section 2 above, the energies of the col-lisions differ slightly ( ∼ 0 . MeV) from those of Ref. [21].As it is not expected that this small difference will signifi-cantly alter the shape of the angular distributions, we haveinterpolated the cross sections of Ref. [21] to the mid-targetenergies and scaled the experimental angular distributionsof Ref. [21] for comparison. The two distributions are pre-sented together in Figure 7 for both energies.Excellent agreement is seen between both distributions.The two-shouldered angular distribution arises from distinctcontributions where only proton and neutron evaporationoccurs, and when 𝛼 evaporation occurs. This structure isvery well reproduced, particularly in the relative magnitudesof the two components, and the position of the secondaryshoulder of the distribution between ◦ .One of the considerable advantages of using SOLI-TAIRE is apparent when we consider that the velocity filtermeasurements (made angle-by-angle) took a number of daysto acquire in full, limited mostly by the time taken to collectenough counts at the largest angles of the distribution. Themeasurements with SOLITAIRE, on the other hand, tookonly tens of minutes. The overall transport efficiency ofSOLITAIRE can be found by comparing the integral of 𝑑𝜎 𝐸𝑅 ∕ 𝑑𝜃 𝑙 𝑑𝑣 𝑙 before and after filtering by the SOLITAIREtransport efficiency. For the S+ Y reaction at both beamenergies, the total transport efficiency, 𝜀 𝑇 , is 92%.The largest discrepancy occurs for the 𝐸 𝑏𝑒𝑎𝑚 = 124 MeVmeasurements at forward angles, where the cross section isunderestimated. This deviation may be a significant reasonfor the discrepancy in the total cross section, discussed in thenext section. At these forward angles, due to blocking by the
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Page 5 of 8 l Laboratory angle, 110 θ Derived DistributionExperimental (vel. fi lter)Distribution Ref [12] E beam = 112 MeV d σ E R /d Ω ( m b / s r) l Laboratory angle, 110 d σ E R /d Ω ( m b / s r) θ Derived DistributionExperimental (vel. fi lter)Distribution Ref [12] E beam = 124 MeV Figure 7:
The differential cross section 𝑑𝜎 𝐸𝑅 ∕ 𝑑 Ω as a functionof laboratory angle, 𝜃 𝑙 , for 𝐸 𝑏𝑒𝑎𝑚 = 112 MeV (top) and 𝐸 𝑏𝑒𝑎𝑚 =124 MeV (bottom). The result of the minimisation routine isshown as a black histogram, while independent experimentaldata [21] is shown in red squares.
Faraday cup, the transport efficiency is particularly low. Interms of the centre-of-momentum velocity distribution, thelargest contributions to these small angles comes from ERswith small recoil velocities 𝑣 𝑐 , and it may be that this portionof 𝑃 ( 𝑣 𝑐 ) is (relatively) poorly constrained by our iterativeprocedure.Whilst the angular distributions are reasonably well de-scribed using two Maxwellians in 𝑃 ( 𝑣 𝑐 ) , it may be that a sumof three Maxwellians (describing 𝑥𝑛 , 𝑝𝑥𝑛 and 𝛼𝑥𝑛 emissionexplicitly) would be more appropriate, with the 𝑝𝑥𝑛 compo-nent, like the 𝛼𝑥𝑛 component, having an offset. Currently,the lower Maxwellian is forced to describe both 𝑥𝑛 and 𝑝𝑥𝑛 emissions, which might lead to an underestimate of small 𝑣 𝑐 . We might expect this problem to be more pronouncedat 𝐸 𝑏𝑒𝑎𝑚 = 124 MeV, where proton emission is more likely,which is consistent with the poorer description of the angu-lar distribution at this energy. The fact that, at both energies,the angular distribution shows a smoother transition betweenthe two shoulders for the velocity filter data, also indicatesthat the ER recoil velocity distribution may be more complexthan assumed. The inclusion of a third Maxwellian wouldcertainly smooth the extracted distribution in this region. At-tempts were made to fit the velocity distribution with three Maxwellians, but there are insufficient counts in the presentdata to unambiguously constrain the parameters, and furthermeasurements are required to clarify this point.A deviation from the independent measurement is alsopresent at larger angles ( 𝜃 𝑙 ≥ ◦ ), where the deduceddistributions underestimate the independent experimentaldata at both energies. This may be due to the presence ofanother emission mode, 𝛼 + 𝑝𝑥𝑛 / Li+ 𝑥𝑛 , which woulddeliver greater recoil to the ERs than the 𝛼 emission,resulting in more widely distributed ERs. Attempts to fit anadditional Maxwellian failed, this time due to inadequateconstraint in the extremes of the velocity distribution whichwould uniquely constrain the Maxwellian contributing tothese largest angles.Despite these discrepancies, the method as it currentlystands is well-suited for measuring ER cross sectionsin systems where neutron evaporation is the dominantemission mode. In such systems, we can expect that theemission is described well by two Maxwellians in thecentre-of-momentum frame. With respect to cross sections, we first consider the to-tal evaporation residue cross sections for the incident beamenergies 𝐸 𝑏𝑒𝑎𝑚 = 112 and 124 MeV, comparing to those ofRef. [21], measured using a velocity filter technique. Due tothe slightly differing energies, we have interpolated the re-sults of [21] to match our mid-target energies. Taking intoaccount measured energy losses in the entrance window car-bon foil, the helium gas prior to the target, and half the targetthickness, our mid-target energy corresponding to 𝐸 𝑏𝑒𝑎𝑚 =112 MeV is 𝐸 𝑙𝑎𝑏 = 111 . MeV, and the extracted evapo-ration residue cross section is 𝜎 𝐸𝑅 = 158 ± 1 mb, wherethe limiting error is the statistical error. This is in excellentagreement with the value of
158 ± 1 mb from Ref. [21].For the 𝐸 𝑏𝑒𝑎𝑚 = 124 MeV runs, the mid-target energy is 𝐸 𝑙𝑎𝑏 = 123 . MeV, and the reconstructed cross section is
424 ± 2 mb. This compares to an interpolated cross sectionof
453 ± 5 mb from Ref. [21]. Possible reasons for thisdiscrepancy in the higher energy data are as discussed in theprevious section. Further, as indicated in the list in Section 2,a number of other quantities need to be precisely known inorder to correctly normalise the experimental distribution,and this discrepancy requires further investigation.
To quantify the uncertainty introduced by the reconstruc-tion routine itself, a test was designed based on statisticalbootstrapping [22]. Sub-samples of the measured data aretaken and run through the routine to calculate the cross sec-tion 𝜎 𝐸𝑅 , exactly as would be done for the total data set. Thedata for these sub-samples is a random sample of events fromthe original data, allowing repeat sampling.By choosing the random sub-samples a number of times(100 has been shown to be sufficient [23]) a mean value ofthe cross section can be calculated, along with the standarderror in the mean for each sub-sample size. The standard L. T. Bezzina et al.:
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Page 6 of 8 fi t424423422421420419418417416 σ E R ( m b ) Bootstrapped mean of σ ER Asymptote of fi t functionFit to bootstrapped mean, f(x) Figure 8:
Results of the statistical bootstrap performed for theroutine described in Section 3. The plot shows mean cross sec-tions calculated from 100 sub-samples at each fractional size,with error bars representing the standard error in the mean, aswell as a fit to these means and the asymptote of that fit. error in the mean is a purely statistical quantity, and esti-mates how far the mean of the sub-samples is likely to befrom the population mean, renormalising the standard devi-ation of the sub-samples by √ 𝑛 , where 𝑛 is the numberof sub-samples. It does not take any systematic error intoaccount. This allows two important properties of the rou-tine to be deduced: its consistency and its bias [24]. If theroutine is consistent, it is expected that the mean value ofthe cross section should converge to the ‘true’ value as thesub-sample size approaches the size of the data set. If theroutine is unbiased , then for any sub-sample size, the rou-tine should always produce a value close to the ‘true’ value,within statistical uncertainties.The resulting mean cross section as a function of sam-ple size is shown in Figure 8. The results show that theroutine is biased at smaller sample sizes, but it is consis-tent, as it asymptotically converges on a value as the sam-ple size increases (approaching a single value in the infinitelimit [25]). The bias is likely introduced due to inadequateconstraint of the second (offset) Maxwellian, since the subsamples tend to have insufficient data away from the cen-tral peak of the velocity distribution. This suggestion is sup-ported by examination of the behaviour of individual param-eter values as a function of sub-sample size. There was somebias apparent in the parameter controlling the width of thefirst Maxwellian, becoming consistent at a sub-sample sizeof 30%. The parameters controlling the width and offset ofthe second Maxwellian, however, showed variations an or-der of magnitude larger, and remained biased at sub-samplessizes up to 50%. As the total cross section is also biased upto these sub-sample sizes, it is suggested that constraint ofthe second Maxwellian is only possible once 50% of the datais used in the routine.In order to test whether our reconstruction routinehad satisfactorily converged to the correct answer, wefit the bootstrapped data with a set of converging func-tions. The chosen set of functions were of the form 𝑐 + 𝑐 𝑥 −1 + ... + 𝑐 𝑛 𝑥 − 𝑛 where the 𝑐 𝑖 are fit parameters. Thesimplest function of this type with a reduced 𝜒 closest to1 (the optimal fit) was 𝑓 ( 𝑥 ) = 𝑐 + 𝑐 𝑥 −1 + 𝑐 𝑥 −2 , addingfurther terms offered no improvement to the reduced 𝜒 ,and so this relatively simple function was chosen. Oncethe fit was performed, the constant parameter 𝑐 could beextracted, which is the horizontal asymptote the function(and hence bootstrapped data) converges to in the infinitelimit (as per our above definition of consistency). Afterfitting, it was found that 𝑐 = 424 .
218 ± 0 . mb.We can then compare our bootstrapped means (redsquares in Figure 8) to the asymptote. When we haveenough data, we would expect that the bootstrapped meanwould lie along the asymptotic line (dashed orange).Currently, the mean values using a sub-sample of 50% orgreater are within ∼ ∼ 𝑣 𝑐 distribution are made, allowing us to assess how wellconstrained a third Maxwellian may be to new datasets withlarger numbers of counts.
5. Summary and Outlook
A new method to characterise the transport efficiencyof the fusion product separator SOLITAIRE has been de-veloped. The approach uses Monte Carlo simulations ofthe transport efficiency and measured ER velocity distribu-tions to iteratively reconstruct the ER angular distributionand cross section. The routine has been benchmarked againstthe reaction of S+ Y at two beam energies: 112 MeV and124 MeV. Good agreement is found between the deduced ERangular distributions and an independently measured angu-lar distribution [21].The cross sections for these reactionshave been determined to be
158 ± 1 mb and
424 ± 2 mb, re-spectively, with a corresponding transport efficiency of 92%for both cases. The total cross section is in excellent agree-ment at 𝐸 𝑏𝑒𝑎𝑚 = 112 MeV. However, it is underestimatedby ∼ 7% at 𝐸 𝑏𝑒𝑎𝑚 = 124 MeV, possibly due to the simpli-fied form of the ER recoil distribution 𝑃 ( 𝑣 𝑐 ) used. This un-certainty will be explored using new cross section measure-ments made using an upgraded version of the SOLITAIREdevices, featuring a new 8T solenoid. L. T. Bezzina et al.:
Preprint submitted to Elsevier
Page 7 of 8 cknowledgements
The project was supported by the Australian ResearchCouncil Discovery Grants FL110100098, DP160101254,DP170102318 and DP170102423. Support for operations ofthe HIAF accelerator facility from the Federal GovernmentNCRIS HIA capability is also acknowledged.
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