MMapping the frontiers of the nuclear mass surface
Zach Meisel
Institute of Nuclear & Particle Physics, Department of Physics & Astronomy, OhioUniversity, Athens, OH, 45701 USAE-mail: [email protected]
October 2019
Abstract.
Nuclear masses play a central role in nuclear astrophysics, significantlyimpacting the origin of the elements and observables used to constrain ultradensematter. A variety of techniques are available to meet this need, varying in theiremphasis on precision and reach from stability. Here I briefly summarize thestatus of and near-future for the time-of-flight magnetic-rigidity (TOF- Bρ ) massmeasurement technique, emphasizing the complementary and interconnectedness withhigher-precision mass measurement methods. This includes of recent examples fromTOF- Bρ mass measurements that map the evolution of nuclear structure acrossthe nuclear landscape and significantly impact the results and interpretation ofastrophysical model calculations. I also forecast expected expansion in the knownnuclear mass surface from future measurement at the Facility for Rare Isotope Beams.
1. Introduction
Nuclear masses, and more specifically nuclear mass differences, are fundamentaldescriptors of atomic nuclei. Mass differences reflect the evolving energetics associatedwith changes in nuclear structure across the nuclear landscape as well as the energycosts (and gains) for nuclear reactions in astrophysical environments. Typical massdifferences of interest are separation energies, e.g. for two-neutrons, S n ( Z, N ) =
M E ( Z, N −
2) + 2
M E (0 , − M E ( Z, N ) (1)and the reaction Q -value Q = Σ reactants M E − Σ products M E, (2)where
M E is the atomic mass excess, Z is the proton number, and N is the neutronnumber. Changes in the slope of S n for neutron-rich isotopes of an element providesignatures of neutron shell and subshell closures, while Q -values are essential inputs intoastrophysics model calculations.Therefore, nuclear mass measurements continue to play an essential role in nuclearphysics studies, in particular for nuclear structure and nuclear astrophysics. Recentcontributions include the emergence of the N = 32 [1] and N = 34 shell closures [2], a r X i v : . [ nu c l - e x ] A p r apping the frontiers of the nuclear mass surface N = 40 [3], Q -value determinations essentialfor calculations of type-I X-ray bursts [4, 5], and determining the trend in masses ofneutron rich nuclei whose imprint can be seen in calculations of astrophysical r -processabundance patterns [6, 7].In all of these cases, precise nuclear mass determinations were required to contributeto solving the problem at hand. However, it is important to note that “precision” is arelative concept, where the necessary mass precision for a given scenario depends on thecontext. For instance, consider the case of Ar( p, γ ), whose dependence on a single low-energy resonance means that keV-level changes to the Q -value leads to tens of percentchanges in the astrophysical reaction rate [8]. Near the other extreme is the much lowerprecision required to constrain the properties of the neutron star crust. As an example,Figure 1 show the equilibrium composition for a cold-catalyzed neutron star crust usingvarious nuclear mass models. Here MeV-level differences between nuclear masses arerequired to modify the onset of the N = 82 shell closure, to which the compositionconverges deep in the outer crust. [MeV] e m Z o r N NZ BSK8FRDM LDMTMA
Figure 1.
Equilibrium neutron number N and proton number Z for depths in a non-accreting neutron star as indicated by the electron chemical potential µ e . Data arefrom References [9] and [10]. The precision that can be achieved in nuclear mass measurement varies forthe available measurement techniques, where the cost of increased precision is oftenincreased measurement time. A map of achieved measurement precision for nuclidesof various half-lives is shown in Figure 3 of Reference [11]. While a nuclear mass m precision of ( δm ) /m < − has been achieved for penning trap mass spectrometry(PTMS), this is in general only possible for a half-life t / > t / apping the frontiers of the nuclear mass surface Bρ ) method canbe used, albeit at the cost of precision. The TOF- Bρ method is generally limited to aprecision of a few times 10 − .The value of a relatively low-precision mass measurement technique can be seenby considering the characteristics of nuclides of interest for astrophysical processes.The difficulty of measuring a nuclide in the laboratory can be quantified using theexoticity [12], E = log (cid:12)(cid:12)(cid:12)(cid:12) dN stab t / ( dN drip + 1) (cid:12)(cid:12)(cid:12)(cid:12) , (3)where dN stab is the number of neutrons from stability along an isotopic chain and dN drip is the same for the neutron dripline, e.g. as defined by the FRDM [13] mass model.Typically (see Figure 6 of Reference [12]), PTMS is limited to E <
1, while TOF- Bρ probes out to roughly E = 4. For context, consider the E distributions for nuclidesinvolved in astrophysical processes shown in Figure 2. It is clear that measurementtechniques accessing E > e N u c l i de s pe r . e e N u c l i de s pe r . e e N u c l i de s pe r . e U nknown Known U nknown
Known U nknown
Known -1 -6 -2 rp -process r -process NS crust Figure 2.
Nuclides involved in astrophysical processes with known (solid-red) orunknown (hatched black) nuclear mass, binned by E . The rp , r , and neutron star crustreaction network paths are from References [14], [15], and [16], respectively. The remainder of this article focuses on the TOF- Bρ mass measurement method.Section 2 briefly summarizes the TOF- Bρ method, Section 3 highlights some significantcontributions of TOF- Bρ measurements to nuclear structure and nuclear astrophysics,and Section 4 provides a preview of mass measurement achievements anticipated at theupcoming Facility for Rare Isotope Beams (FRIB). apping the frontiers of the nuclear mass surface
2. The TOF- Bρ mass measurement method The concept for the TOF- Bρ method is that the nuclear mass can be determined byequating the centripetal and Lorentz forces on a charged massive particle (i.e. a nucleus)moving through a magnetic system. After some straightforward algebra and applyinga relativistic correction in the form of the Lorentz factor γ , it is apparent that the restmass m = TOF L path qBργ , (4)where TOF is the time-of-flight along a path of length L path for an ion with charge q andmagnetic rigidity Bρ . In practice, this relationship is not practicable for nuclear massdeterminations of the necessary precision. For instance, for the typical conditions of aTOF- Bρ experiment [17], using Equation 4 to determine a nuclear mass to the 10 − -level would require knowing L path ∼
60 m to tens of microns. Instead, an empiricalrelation is established by determining the Bρ -corrected TOF, TOF (cid:48) , of several nuclideswhose mass is known, e.g. from PTMS, to high-precision: m = f ( Z, A = Z + N, TOF (cid:48) ) . (5)Ultimately, the empirical approach requires determining the average TOF withinless than a picosecond and Bρ via a sub-millimeter measurement of the verticaldisplacement at a dispersive focus for several tens of nuclides. Ideally, the nuclideswhose mass is known (“reference nuclides”) have Z and A/Z similar to the nuclides ofinterest.Uncertainty quantification in the TOF- Bρ method provides a significant challenge,due to the careful consideration required to assign systematic uncertainties. To aid inthis discussion, consider the Rumsfeld Quadrant [18] shown in Figure 3. Known Knowns Unknown KnownsKnown Unknowns Unknown Unknowns
Calibration Masses Systematic Spread inFit ResidualsUncertainty in Fit Parameters Uncertainty in Fit Function
Figure 3.
TOF- Bρ mass measurement uncertainty considerations categorized via aRumsfeld Quadrant, following Reference [18]. apping the frontiers of the nuclear mass surface ∼
500 counts). An unknown that we knowof is the systematic spread remaining in the residuals of the fit to reference nuclideswhich nearly always have χ ν >
1. This uncertainty is generally accounted for by addinga blanket uncertainty in m/q (as the actual fit function used determines this quantity)until χ ν = 1. The second, and unfortunately often omitted, systematic uncertaintycomes from the unknown unknown: we do not know if the relation ultimately used inEquation 5 is the fit-function that best describes the data. While Occam’s razor dictatesthat the simplest model should be preferred, this simplicity needs to be balanced withthe quality of the overall fit. A suggested approach is to use ∆ χ = χ i − χ , where χ i applies to a given model and χ is the model resulting in the overall best-fit. Theset of fit-functions which nominally describe the data equally well within some degreeof confidence can be determined using ∆ χ tabulated for the number of degrees offreedom [19].
3. TOF- Bρ contributions to nuclear structure and nuclear astrophysics
10 20 30 40 50 60 70 80 N Z NSCL SPEG TOFI RIKEN
Figure 4.
Nuclear masses measured by the TOF- Bρ method at the NSCL [20], SPEGat GANIL [21], TOFI at LAMPF [22], and RIKEN [2]. More than 300 nuclear masses have been determined using the TOF- Bρ method, apping the frontiers of the nuclear mass surface Bρ has been employed at NSCL [20] and RIKEN [2]. The concentrationof measurements for Z <
30 is largely due to the difficulty of dealing with multiplecharge-states present for studies of higher- Z nuclides, though efforts are ongoing toaddress this difficulty, e.g. using the technique of Reference [23].Nuclear structure studies have identified regions of shape coexistence, theappearance and disappearance of shell closures, and the existence of halo nuclides. Forinstance, the lower-bound of the N = 28 shell closure was mapped by References [24,25, 26]. The properties of halo nuclides were determined by References [22, 27], wherethe latter was key to establishing the two-neutron halo nature of C and one-neutronhalo of Ne.Achievements in nuclear astrophysics have largely focused on improving modelsof the accreted neutron star crust, where nuclear masses determine the location andstrength of heat sources and heat sinks occurring due to electron-capture reactions [9].Thus far, measurement results [17, 28] have indicated that electron-capture heat sourcesappear to be weaker than previously predicted. However, the accreted neutron star crustis not as cool as was once thought possible, since Reference [29] found that the strongestpredicted heat sink in fact does not exist. Work is ongoing to expand TOF- Bρ studiesof astrophysical interest to the r -process region [30].
4. The future of TOF- Bρ mass measurements While a handful of measurement targets remain at existing facilities, to significantlyextend the TOF- Bρ method to more exotic isotopes will require state-of-the-artradioactive ion beam facilities, such as FRIB. The purpose of this section is to forecastthe extent to which the known nuclear mass surface is likely to be expanded at FRIB bycoordinated efforts in TOF- Bρ and PTMS measurements. This is done by discussingpredicted FRIB production rates followed by anticipated achievements in PTMS, TOF- Bρ , and the two techniques combined. FRIB production rate predictions ‡ are calculated using the software from Reference [31],whose assumptions are briefly described here. Fast-beam rates, required for TOF-B ρ ,were calculated employing the KTUY mass model [32], EPAX 2.15 fragmentation crosssection parameterization [33], LISE++3EER model for production cross sections fromin-flight fission [34], and LISE++v9.2.68 for beam transmission efficiency [35]. In eachcase the rate chosen is that from the optimum primary beam, i.e. the one of the 47 ‡ At present, calculations for individual nuclides can be obtained at https://groups.nscl.msu.edu/frib/rates/fribrates.html. apping the frontiers of the nuclear mass surface ∼ particles per second (pps) are anticipated for nuclidesjust beyond the present limit of known masses, where the production rate generallydrops off one order of magnitude for every 1-2 additional neutrons from stability. Thestopped-beam rate is 10% of the fast-beam rate on average, but mostly ranges from0-40% (0% cases are due to short half-lives). PTMS, the highest-precision mass measurement technique presently available for rareisotopes, will be performed with the Low Energy Beam Ion Trap (LEBIT) Penning trapat FRIB [36]. The PTMS technique consists of obtaining nuclear masses by measuringthe resonant frequency of the nucleus of interest with respect to the resonant frequencyfor an ion (or typically atomic cluster) of known mass orbiting within a few cubic-centimeter volume, confined by a strong magnetic field and hyperbolic electrodes [37].PTMS has been demonstrated to deliver a mass measurement precision of 10 − orbetter for as little as ∼
50 measured ions [38] and for nuclides with half-lives as short as ∼
10 ms [39].The time-of-flight ion-cyclotron-resonance (TOF-ICR) technique is predominantlyemployed for PTMS. For TOF-ICR PTMS, the cyclotron resonance of the ion in the trap,which is directly proportional to its mass, is identified by converting the orbital motionin the trap to a longitudinal motion out of the trap and finding the minimum TOFto a fixed detection location. The measurement uncertainty for TOF-ICR is reducedby storing individual ions for long times in the trap and observing several ions over alarge enough frequency range to map the cyclotron resonance. The relative statisticaluncertainty δm/m , which is generally much larger than the systematic uncertainty forPTMS of rare isotopes, is roughly given by δm/m ≈ R − n − / , where n is the number ofions detected and R is the resolving power [40]. The resolving power is approximatelyequal to the product of the cyclotron frequency of the ion in the trap f c (typically O ∼
MHz) and the length of time the ion orbits in the trap t obs (typically O ∼ R therefore depends on many considerations, such as the mass of the nucleus of interest,the obtainable charge-state, the time it takes to produce the optimum charge state, andthe nuclear half-life.Given the uncertainties in charge-breeding capabilities and the approximate natureof the estimate for n, I make the approximation that R = 10 for all nuclides ofinterest, which is in-line with sample cases for rare isotopes [41]. For simplicity, Iassume t obs =100 ms, and therefore n will be the product of the stopped-beam rate apping the frontiers of the nuclear mass surface β -decay half-lives from Reference [42] are used whenavailable and predictions from Reference [43] are used otherwise. I assume a systematicuncertainty typical for the measurement precision of reference ions, δm/m = 10 − [44].Above A ≈
50, the known mass surface will be extended by PTMS by a fewisotopes or more for each isotopic chain. The improvement over current nuclear massuncertainties is also significant, considering that many nuclear masses at the presentexperimental frontier are only known to precisions of δm/m ≈ − [45]. The greatestgains are expected for neutron-rich isotopes with Z >
55, which will substantiallyimprove the predictive power of rare-earth element nucleosynthesis in the r -process [46].Note that the predicted gains on the proton-rich side are unreliable, as particle-decaysare not taken into account in our estimates. Furthermore, the estimates neglect thepotential existence of isomeric states and isobaric contaminants [47]; however, recentimprovements in PTMS, such as the stored waveform inverse fourier transform [48]and phase-imaging ion-cyclotron-resonance [49] techniques will mitigate the impactof these complications. Additionally, the reach of PTMS may be extended by near-future developments such as the single ion penning trap (SIPT) method [44]; however,expectations for SIPT have yet to be benchmarked with rare isotope measurements andso are not considered here. Bρ mass measurement The TOF- Bρ method is described in Section 2. The details of the uncertainty evaluationare elaborated upon here as they are pertinent to developing a forecast of anticipatedmeasurement results. It is important to note that the TOF- Bρ method relies on theavailability of PTMS results for nuclides nearby (in terms of Z and N ) the isotopes ofinterest and is primarily used as a tool to extend the known mass surface by a few moreneutrons along an isotopic chain.Unlike PTMS, the measurement uncertainty of TOF- Bρ is generally dominated bysystematic uncertainties due the many unknowns which must be accounted for alongthe large experimental set-ups [17, 40]. Therefore, the estimation technique for themass measurement uncertainty achievable via TOF- Bρ is somewhat more approximate.The statistical uncertainty of TOF- Bρ is related to the TOF measurement precision σ TOF /TOF and number of measured ions n by δm/m ≈ σ TOF / (TOF √ n), wherea typical σ TOF / TOF of 10 − [50, 28, 26] is assumed here. I base the systematicuncertainty on the rough empirically motivated approximation § [17, 26, 27, 28, 29, 50]that δm/m | syst = 5 × − (1 + ( N − N ref )), where N − N ref is the number of neutronsseparating the nuclide of interest and the most neutron-rich isotope of that element with § Note that the true systematic uncertainty depends on many factors, not least the details of the localTOF-mass relationship and the availability of suitable reference nuclides [17]. apping the frontiers of the nuclear mass surface
9a mass uncertainty ≤ − . I sum the statistical and systematic uncertainties to arriveat a total uncertainty and assume a measurement time of 100 hours, as is typical forrecent TOF- Bρ experiments.In general TOF- Bρ extends the measurable mass surface by 1-3 nuclides witha precision useful to applications in nuclear structure and nuclear astrophysics ( (cid:46) a few times 10 − ). The given estimates ignore potential complications such as theexistence of isomers, multiple charge-states, and magnetic rigidity limits of experimentalequipment [40]; however, the former will generally require further experimental work,techniques to deal with multiple charge states [23] have recently been implemented forTOF- Bρ [30], and it is anticipated that FRIB will host the high-rigidity spectrometerwith a more than sufficient maximum rigidity, so I do not consider these complicationsfurther. N Z -8 -7 -6 -5 -4 Valley of stabilityKnown mass in AME 2012r-process pathrp-process pathFRDM 2n-drip-line
Figure 5.
Anticipated relative uncertainty δm/m achievable by the combined use ofPTMS and the TOF- Bρ method at FRIB. The rp and r -process paths are the same asthose referred to in Figure 2. Note that life-time reductions due to particle emissionare neglected, and therefore estimates near the proton drip-line are not reliable. Figure 5 shows the predicted mass measurement precision that can be achieved by thecombined use of PTMS and TOF- Bρ at FRIB. Relative to the 1,098 neutron-rich massesreported in the 2012 Atomic Mass Evaluation [45], the predictions shown correspond to965 higher-precision nuclear masses and 1,172 new nuclear masses on the neutron-richside of stability: 693 from PTMS with a precision ≤ − and 479 from TOF- Bρ with a apping the frontiers of the nuclear mass surface ≤ − . These measurements would roughly double the known mass surface forneutron-rich nuclides, leading to advances in nuclear structure and nuclear astrophysics.Masses will likely be obtained very near to the neutron drip-line up to roughly iron andwill elucidate the evolution of the N = 82 and (especially) N = 126 shell-closures fordecreasing proton numbers. The expansion of the mass surface up to A ≈
100 willtightly constrain the possible strength of nuclear heating and cooling in the crusts ofaccreting neutron stars [9]. Whereas the expansion for the isotopes above iron maydeliver nuclear masses along the majority of the r -process path, possibly distinguishingbetween hot and cold r -process sites, for example with the neodymium masses [51].
5. Conclusions
The TOF- Bρ method has played and will continue to play a key role in massspectrometry for exotic nuclides. Over the past three decades, such measurements havemade significant contributions to our understanding of nuclear structure and nuclearastrophysics, particularly invovling neutron-rich nuclides. Though lower precision thanother available methods, TOF- Bρ measurements continue to map the frontiers of thenuclear mass surface. Acknowledgments
I thank my many TOF- Bρ mass measurement collaborators, especially SebastianGeorge, Alfredo Estrad´e, Mike Famiano, Wolfi Mittig, Fernando Montes, HendrikSchatz, and Dan Shapira. This work was supported in part by the U.S. Departmentof Energy Office of Science under Grants No. DE-FG02-88ER40387 and DESC0019042and the U.S. National Science Foundation through Grant No. PHY-1430152 (JointInstitute for Nuclear Astrophysics – Center for the Evolution of the Elements). References [1] Leistenschneider E et al.
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