The Isotopic Abundances of Galactic Cosmic Rays with Atomic Number 29 ≤ Z ≤ 38

The Cosmic Ray Isotope Spectrometer (CRIS) on the Advanced Composition Explorer spacecraft has been operating successfully in a halo orbit about the L1 Lagrange point since late 1997. We report here the isotopic composition of the Galactic cosmic ray (GCR) elements with 29 ≤ Z ≤ 38 derived from more than 20 years of CRIS data. Using a model of cosmic-ray transport in the Galaxy and the solar system (SS), we have derived from these observations the isotopic composition of the accelerated material at the GCR source (GCRS). Comparison of the isotopic fractions of these elements in the GCRS with corresponding fractions in the solar system gives no indication of GCRS enrichment in r-process isotopes. Since a large fraction of core-collapse supernovae (CCSNe) occur in OB associations, the fact that GCRs do not contain enhanced abundances of r-process nuclides indicates that CCSNe are not the principal source of lighter (Z ≤ 38) r-process nuclides in the solar system. This conclusion supports recent work that points to binary neutron-star mergers, rather than supernovae, as the principal source of galactic r-process isotopes.


Introduction
The abundances of the isotopes and elements in Galactic cosmic rays (GCRs) provide important constraints on their origin. The Cosmic Ray Isotope Spectrometer (CRIS; Stone et al. 1998b) on the Advanced Composition Explorer spacecraft (ACE; Stone et al. 1998a), which was launched on 1997 August 25 into a halo orbit around the L1 Lagrange point, was designed to measure the elemental and isotopic abundances of nuclei with atomic number, Z, in the range 4 Z 30. In several papers (e.g., Wiedenbeck et al. 1999Wiedenbeck et al. , 2007Binns et al. 2005Binns et al. , 2016 the excellent isotopic resolution and statistics achieved by CRIS for Z 28 have been demonstrated.
The relative abundances of nuclei with Z > 28 decrease dramatically with increasing Z; for example, the abundances in the solar system (Lodders et al. 2009) of 30 Zn and 32 Ge are lower than that of 28 Ni by factors of ∼40 and ∼400, respectively. Even though the CRIS geometrical factor of ∼250 cm 2 sr is larger than that of prior instruments that measured iron-group cosmic-ray isotopes (Leske et al. 1992;Connell & Simpson 1997;Lukasiak et al. 1997;DuVernois 1997), it was not anticipated that CRIS would be able to measure isotopes with Z > 30 in its planned lifetime requirement of 2 yr and goal of 5 yr. As it turns out, the ACE spacecraft and the CRIS instrument are still operating effectively as this paper goes to press, after more than 24 yr in space. This long exposure time of CRIS, coupled with its large geometrical factor, has enabled accumulation of sufficient numbers of events to measure the isotopic composition of 31 Ga, 32 Ge, 34 Se, 36 Kr, and 38 Sr for the first time, as well as achieving low-statistics indications of the isotopic composition of 33 As and 35 Br. In addition, we have obtained measurements of 29 Cu and 30 Zn isotopes with greatly improved statistics over those previously published based on CRIS data from the early part of the ACE mission (George et al. 1999(George et al. , 2000. Earlier isotopic-composition measurements from CRIS have given valuable insights into the origin of cosmic rays. CRIS has shown that the abundance in the cosmic rays of 59 Ni, which decays by electron capture with a half-life 7.6 × 10 4 yr, is consistent with the absence of this nuclide in cosmic-ray source material, indicating a delay between nucleosynthesis and cosmic-ray acceleration of 10 5 yr (Wiedenbeck et al. 1999; also see Neronov & Meynet 2016 for an opposing view). The discovery in cosmic rays by CRIS of the radioactive isotope 60 Fe, which decays by β − emission with a half-life 2.6 Myr, and the measurement of the 60 Fe/ 56 Fe ratio indicate that the time required for acceleration and transport to Earth does not greatly exceed the 60 Fe half-life and that the 60 Fe source distance does not greatly exceed 1 kpc, consistent with an origin in a nearby cluster of massive stars ). In addition, it has been shown that the ratio 22 Ne/ 20 Ne in the cosmic rays is ∼5× larger than in the solar system (Binns et al. 2005, and references therein). This 22 Ne enrichment gives support to models of cosmic-ray origin in OB associations, where normal interstellar material similar in composition to the solar system is enriched by outflow from massive stars, in particular Wolf-Rayet stars (Higdon & Lingenfelter 2003;Binns et al. 2005Binns et al. , 2008, and references therein).
In this paper we present measurements of the relative abundances of the isotopes of Cu, Zn, Ga, Ge, As, Se, Br, Kr, and Sr over the energy range of ∼130 to 700 MeV/nucleon. Using those observed abundances and a model of cosmic-ray transport in the Galaxy and the solar system, we derive the relative abundances of these isotopes at the Galactic cosmic-ray source (GCRS). Comparison of these GCRS isotopic abundances with corresponding abundances in the solar system (Lodders et al. 2009) demonstrates that the interstellar medium (ISM) in the region where the cosmic rays originate has composition similar to, but not identical with, that of the solar system.
In Section 2 we describe how the CRIS instrument determines nuclear mass. In Section 3 we present the isotope abundances observed by CRIS. In Section 4 we introduce the instrumental corrections that are needed to derive abundances incident on the instrument from the measured numbers of events. In Section 5 we derive the abundances at the cosmic-ray source from the measured abundances using a model of transport in the Galaxy and the solar system. In Section 6 we discuss these results.

Isotope Data
The CRIS instrument, which consists of four stacks of silicon detectors and a scintillating optical fiber hodoscope, is illustrated in Figure 1, where only two of the four silicon stacks are shown. Each stack contains 15 silicon wafers, each of 10 cm diameter and of 3 mm thickness, giving a total siliconstack thickness of 4.5 cm, in which multiple measurements of energy loss versus residual energy for stopping nuclei are made. The scintillating-fiber hodoscope is used to measure the trajectories of detected particles, which, in conjunction with detector thickness maps, enables the accurate determination of path lengths through the detectors that are penetrated. The energy loss (ΔE) versus residual energy (E¢) method is used to determine particle charge and mass (Stone et al. 1998b(Stone et al. , 1998c. The 15 silicon wafers are grouped together to form nine detectors (E1 through E9), as shown in Figure 1, each of which makes a measurement of energy deposition for particles penetrating or stopping in it. Particles stopping in a detector EN are referred to as "range-N" events. So, for example, a particle stopping in detector E6 is referred to as a range-6 event. The instrument is described in detail by Stone et al. (1998b).  . Histogram of nuclei detected by CRIS on a "pseudo-Z" scale with charge (Z) from Cu through Sr. Calculated locations of stable isotopes are indicated by "+" symbols and those of isotopes that decay exclusively by electron capture by filled circles.
In Figure 2 we show a histogram of the "pseudo-Z" or "calculated charge" for nuclei from Cu through Sr. The pseudo-Z depends on both the atomic number (Z) and the atomic mass (A) of the nucleus. The identity of the peaks can be confirmed through a mass calculation, but the results are presented most easily in terms of "pseudo-Z" derived as follows. On a plot of ΔE versus E¢, particles of a given Z and A form approximately parallel response "tracks," with relatively wide spacing between adjacent elements and narrower spacing between isotopes of a given element. Examples are shown in Stone et al. (1998bStone et al. ( , 1998c. The approximate track locations can be calculated using a range-energy relation for heavy nuclei in silicon (Stone et al. 1998c). Assuming a nominal A/Z ratio, a noninteger Z value (which we call "pseudo-Z") can be assigned to each event, with different isotopes of the element separated by fractional numbers of charge units. To calibrate the relationship between pseudo-Z and isotope mass, we proceed as follows. Using nominal values for the thicknesses of the ΔE and E¢ detectors and the dead layer between them, and also for the mean angle of incidence of the particles relative to the detector normal, we employ the silicon range-energy relation to calculate ΔE and E¢ for particles of a given Z and A, stopping at different depths in the E¢ detector. For each ΔE-E¢ pair, we derive pseudo-Z and then average over all of the calculated cases. In the resulting histogram of pseudo-Z values, elements appear as groups of peaks, with each peak corresponding to an individual isotope, as shown in Figure 2. We see that there are clear peaks for the most abundant stable isotopes of each element up through Ge, and all of the elements are cleanly separated. The "+" symbols above each element in the histogram designate the calculated positions in pseudo-Z space of the stable isotopes for each element. The solid circles represent isotopes that are radioactive and decay only by electron capture. We note that the correspondence between the calculated positions of the isotopes and the histogram peaks is quite good for elements with sufficient numbers of particles so that peaks are readily apparent (i.e., Cu through Ge).
Specifically, to obtain the scaling factor to go from pseudo-Z units (pzu) to mass units (amu), we fit the Cu isotopes in pseudo-Z space and find that they are separated by 0.2097 pzu. Since the two stable isotopes of Cu are separated by 2 amu, we thus have a scaling factor 9.537 amu/pzu. Then, for each element, we pick a reference isotope and translate the mass histogram so that it lines up with the integral mass of that reference isotope. Since the statistical sample of the Cu isotopes is greater than for the heavier elements, we have used this scaling factor for all charges. In Figure 2 we see that using this scaling factor and translation, the peaks of Zn and Ga line up nicely on integral-mass isotopes. For higher-Z the best-fit peaks drift off of the integral-mass numbers. 5 The statistical samples for As, Se, Br, Kr, and Sr have fewer particles, but it is still possible to obtain useful information from them.
Particles analyzed in this study were required to stop in detectors E2 through E8 and have incidence angles to the detector normal of 45°. The CRIS detectors (Allbritton et al. 1996) were fabricated from lithium-drifted silicon, Si(Li), which results in particularly thick dead layers on one surface, with typical thicknesses ∼60 μm and sometimes exceeding 100 μm. Some of the energy lost in these dead layers does not contribute to the measured signal, causing a distortion of the shape of response tracks. To minimize the effects of dead layers on the isotope resolution, we do an approximate calculation of the depth at which each particle stops in the stack and reject those that could be significantly affected. Specifically, particles stopping in detector E2 within 25 μm of the top surface and 450 μm of the bottom surface were rejected. Particles stopping in detectors E3 through E8 within 25 μm of their top surfaces and 150 μm of their bottom surfaces were rejected. 6 The numerical values used for these cuts were determined empirically by examining the shapes of the response tracks near the end of a particle's range. In particular, near the end of a particle's range the cuts were chosen to eliminate "foldback" events, i.e., particles that stop in the top or bottom dead layers of the silicon detectors. These events produce distribution tails that could result in misidentification of the mass of a particle as a lower-mass nucleus. An examination of the lower edges of the distributions in Figure 2 shows that there are no significant tails produced by foldback events on the mass distributions.
We employed multiple approaches for rejecting nuclei that underwent nuclear interactions in the active material of the instrument or in inactive material surrounding it. Many nuclei interacting in the detector stacks are indicated by signals in either the penetration counter (E9) or the active guard counters (G2 through G7) that surround the main active areas of detectors E2 through E7 (Stone et al. 1998b). This is a result of the fact that in most nuclear interactions of a high-Z nucleus within the instrument, lower-charged spallation products of the interaction, such as protons or alpha-particles, will penetrate well beyond the primary particle, usually into the penetration counter or out of the sides into the guard counters.
For events that do not include a signal in any penetration or guard counter, 7 we compare the pattern of energies deposited in the detectors with those expected for a heavy nucleus entering the detector stack through the E1 detector and coming to rest between E2 and E8. Specifically, we require consistency in the charge that is estimated using different detectors for the ΔE measurement. For each event, we make N − 1 estimates of charge, Z, where N is the detector (or range) in which the particle stops (see Figure 1). For a particle stopping in the Nth detector, these N − 1 charge estimates each use one of the detectors E1 through E(N − 1) for ΔE and the sum of detectors following the ΔE detector through detector N as E¢. We require that the rms deviation of the multiple charge estimates from the mean be less than 0.5 charge units. This selection is expected to remove most of the interacted events not eliminated by the penetration counter or the guard counters. Nuclei stopping in E2 have only a single charge estimate, so we rely entirely on the penetration and guard counters to reject interactions for those events.
We also examine the incident particle trajectory derived using the signals in the fiducial areas of the six hodoscope planes (H1X through H3Y in Figure 1) to assure that the particle entered within the geometrical acceptance defined for the instrument and produced a hit pattern consistent with a straight-line, heavy-ion track. Particles that did not meet these criteria were rejected. This rejection of incorrect trajectories is particularly important because if an erroneous value is used for the incidence angle, all of the individual mass determinations are shifted by approximately the same factor, thus nullifying the usefulness of a cut based on the rms deviation among them. The effect of these cuts on detection efficiency is addressed in Section 4.2 and Appendix A.2.
The data were collected over the time period from 1997 December 4 through 2019 February 18. In general, the instrument operated continuously with the exception of occasional, extreme solar active periods, when CRIS automatically turned off the hodoscope high-voltage supplies (George et al. 2009). Although smaller solar particle events with extreme enrichment in ultraheavy elements have been observed (Mason et al. 2004), the energy spectra in such events are too soft to be detected at CRIS energies, and therefore no additional quiet-time selection was necessary. The number of days of actual data collection over this time period was 7406.

Isotope Abundance Measurements
In this section we present the measured numbers of events of cosmic-ray isotopes for elements with 29 Z 38. For each of these elements, a mass histogram of observed events is presented in Figure 3. From these histograms we derive the abundances at the top of our instrument by making several instrument corrections, described in Section 4 and Appendix A. (These measured numbers of nuclei correspond to 1.54 × 10 6 Fe and 7.90 × 10 4 Ni nuclei.) The histograms for elements with sufficient statistics have been fit using a Gaussian maximumlikelihood (GML) program (Scott 2020) using the method discussed by Orear (1982). The Gaussian standard deviation is a free parameter in the fit and is constrained to be the same for all isotopes of the element being fit. In addition, the fit requires that the isotope peaks of a given element be an identical number of mass units apart, but does not require that they fall at integer mass numbers. For elements with Z 32 the GML fit is used to obtain isotope abundances and the statistical uncertainty is that determined by the fit, which takes Poisson counting statistics into account.
For Cu and Zn, the elements with the largest statistical samples, several other approaches were also tried for determining the number of counts due to each isotope. Besides a simple GML fit, a version was tried in which tails of the peaks were not used in the fit, and another in which a low-level background was subtracted from the histogram before fitting. In these cases, averages of results obtained using the different approaches yielded the numbers of observed counts reported in Table 1 (column 3), while the spread among values obtained with different approaches was the basis for assigning systematic uncertainties (column 6). For elements with Z > 30 no systematic uncertainties were included because they would be much smaller than the statistical uncertainties.
For elements with Z > 32 the small statistical samples did not justify peak fitting, so simple mass cuts were used and the statistical uncertainty was determined using Poisson statistics (Gehrels 1986). Table 1 gives in column 3 the observed number of counts. Columns 4 and 5 give the statistical uncertainties, and the estimated systematic uncertainties are in column 6. Columns 7 and 8 give the total measurement uncertainties, which are taken to be quadratic sums of the statistical uncertainties with the estimated systematic uncertainty.
In addition to the isotope abundances given in Table 1, we also report element abundances in Appendix B and Table 5.

29 Cu
The only stable Cu isotopes are 63 Cu and 65 Cu. In Figure 3(a) we show a histogram of the Cu isotopes. Using a GML fit to the gray area of the histogram, we assign 511 and 204 particles, respectively, to 63 Cu and 65 Cu. The fit is overlaid with the histogram. In addition to the particles in this mass range, there are a small number of particles on both the low and high sides of the distributions (see "+" symbols in Figure 3(a)).

30 Zn
Zn has five stable isotopes, 64 Zn, 66 Zn, 67 Zn, 68 Zn, and 70 Zn. In Figure 3(b) we show a histogram of the Zn isotopes. There are three clearly resolved isotopes: 64 Zn, 66 Zn, and 68 Zn. In addition to those isotopes, 67 Zn and 70 Zn are stable isotopes with low fractional abundances in the solar system (Lodders et al. 2009). We have fit the gray area of the histogram with the GML-fitting routine. As was the case for Cu, there are a small number of events on the low side of the distribution not included in the fit that are plotted as "+" symbols. Note that the GML fit obtains finite abundances for A = 67 and 70.

31 Ga
Ga has just two stable isotopes, 69 Ga and 71 Ga, and one isotope, 67 Ga, that decays exclusively by electron capture. In the cosmic-ray source, short-lived electron-capture nuclides such as 67 Ga are absent, but can be produced during Galactic propagation at high energies by fragmentation of heavier cosmic rays. Typically, these fragmentation products are produced with no atomic electrons attached and are effectively stable. Thus, we treat pure electron-capture isotopes as absent in the cosmic-ray source, but take account of electroncapture isotope production and decays that can occur during transport in the Galaxy (see Section 5). Figure 3(c) presents a histogram of the Ga isotopes. We measure a finite abundance of 67 Ga in the arriving cosmic rays. Any systematic uncertainties for Z 31 are believed to be negligible compared to the statistical uncertainties, so are not included. As a check on the fit abundances we have compared the number of particles assigned to each mass by making cuts shown on the figure (see vertical dashed lines). We find that for 67 Ga, 69 Ga, and 71 Ga the numbers of particles obtained by cutting are 4, 35, and 20, respectively. Comparing these with the GML fit counts in Table 1, we see that there is excellent agreement.

32 Ge
Ge has five stable isotopes, 70 Ge, 72 Ge, 73 Ge, 74 Ge, and 76 Ge, and two, 68 Ge and 71 Ge, that decay exclusively by electron capture. Figure 3(d) shows a histogram of the Ge isotopes. We see that the fit obtains finite abundances at 71 Ge and 73 Ge in addition to the clear peaks at 70 Ge, 72 Ge, and 74 Ge, plus the few particles at 76 Ge.

33 As
The element As has only one stable isotope, 75 As, and one, 73 As, that decays exclusively by electron capture; all other isotopes are unstable to β − or β + decay with short half-lives. Figure 3(e) is a histogram of the As isotopes. We have Figure 3. Mass histograms of individual elements 29 Cu through 38 Sr. The curves superimposed on the figures are obtained using a Gaussian maximum-likelihood fitting routine. Vertical dashed lines indicate boundaries adopted when using particle counting to determine relative abundances. These lines are discussed in the text. For Z 34 (Se) the low numbers of particles do not allow us to obtain abundances for individual isotopes, but we can obtain abundances for s-and r-process groups of isotopes. The bin width for all of the plots is 0.2 amu, with the exception of Ga (panel c), which has a bin width of 0.125 amu. estimated the isotopic abundances using a simple cut at A = 74 (vertical dashed line) since the estimated particle masses are well separated and the number of particles is so small. The number of particles assigned to 73 As is four and the number for 75 As is seven. The single particle with A ∼ 71.7 is assumed to be a background count and is ignored.

34 Se
Se has six stable isotopes, 74 Se, 76 Se, 77 Se, 78 Se, 80 Se, and 82 Se, and two, 72 Se and 75 Se, that decay exclusively by electron capture. Figure 3(f) is a histogram of the Se isotopes. The mass resolution combined with the low numbers of particles make it difficult to obtain individual isotope abundances. However, there is a clear break at A = 79 separating the events with A < 79 from those with A > 79. This is expected since 79 Se decays rapidly by β − emission. The events for A < 79 are mostly s-process nuclei, while those with A > 79 are almost entirely r-process (West & Heger 2013;Prantzos et al. 2020), so rather than using the fit results for individual isotopes, we make a cut at A = 79 (vertical dashed line) and examine the two isotope groups. We note that there are zero particles identified as 72 Se.

35 Br
Br has two stable isotopes, 79 Br and 81 Br. Figure 3(g) is a histogram of the Br isotopes. The statistical sample is very small, consisting of only nine particles. We have estimated the isotopic abundances using a simple cut at A = 80 (vertical dashed line) since the events are well separated and the number of events is so small.

36 Kr
Kr has six stable isotopes, 78 Kr, 80 Kr, 82 Kr, 83 Kr, 84 Kr,and 86 Kr. In addition, 81 Kr decays only by electron capture. Figure 3(h) is a histogram of the Kr isotopes. Similar to Se, the lighter nuclides (A < 83) are entirely s-process, while those that are heavier (A > 83) are mostly r-process (West & Heger 2013;Prantzos et al. 2020). The statistics are too poor to justify using the fit results for individual isotopes. However, there is a clear  break between particles with A > 83 and those with A < 83. So we make a cut at A = 83 and examine the two isotope groups.

37 Rb
Rb has two essentially stable isotopes, 85 Rb, which is strictly stable, and 87 Rb, which has a β-decay half-life ∼5 × 10 10 yr and is effectively stable on the timescale of cosmic-ray confinement in the Galaxy. In addition one isotope, 83 Rb, decays only by electron capture. However, there are only seven counts total. Figure 3(i) is a histogram of the counts. Since there is no clear clustering of events, we have chosen to normalize the distribution so that the two highest-mass counts have been assigned to A = 87. Because of the small number of counts that are not nicely grouped, we have not attempted to assign abundances.

38 Sr
Sr has four stable isotopes, 84 Sr, 86 Sr, 87 Sr, and 88 Sr. In addition, there are two isotopes, 82 Sr and 85 Sr, that decay only by electron capture. Figure 3(j) is a histogram of the Sr isotopes. We see a distinct peak at A = 88 and a scattering of particles with lower mass. Again the statistics are too poor for A < 88 to justify using those fit abundances for individual isotopes. So we divide the particles into two groups, those with A < 87 and those with A > 87. According to West & Heger (2013), the Sr isotopes are all s-process nuclei, and, according to Prantzos et al. (2020), they are nearly all s-process (see Table 3, columns 15 and 16).

Mass Resolution
As was mentioned above, the GML fit assumes that the resolution in mass is identically the same for all isotopes of a given element. Figure 4 shows the mass resolution obtained by the GML fits versus atomic number, Z. The error bars are the uncertainty in the mass resolution obtained by the fit. This figure shows that the resolution in mass is in the neighborhood of 0.4 amu for all of these elements.

Corrections to Measurements
We have made three corrections to the number of events of each isotope detected by CRIS to obtain abundances in space (i.e., entering the instrument). They are corrections for (1) particles producing detector signals so large that they saturate one or more of the CRIS pulse-height analyzers (PHAs), (2) loss of particles due to nuclear interactions within the instrument, and (3) small differences in the energy intervals over which different isotopes stop in the detector stack.

Pulse-height Analyzer Saturation
The CRIS PHAs have dynamic range suitable for the original CRIS objective of measuring nuclei with Z 30, but for some nuclei with Z > 32 some of the PHAs saturate. Events with any saturated PHA had to be rejected, leading to reduced detection efficiency that depends on Z, A, angle, and energy. Derivation of the resulting efficiency is described in Appendix A.1. Of the three corrections described here, this PHA-saturation correction is the most steeply dependent on Z.

Nuclear Interactions Within CRIS
The rejection of events due to nuclear interactions in the instrument resulted in a reduction of detection efficiency dependent on Z, A, and energy. We derived this efficiency using Westfall et al. (1979) cross sections, as described in Appendix A.2.

Energy Intervals
CRIS determines both Z and A for incident nuclei that stop in one of the detectors E2 through E8. The energy interval for stopping depends on Z and A. We introduce a correction factor for each Z and A so that those results correspond to the same energy interval for all nuclides. Appendix A.3 describes the derivation of this correction factor.

Cosmic-ray Source Composition
In order to derive the composition of the material that has been accelerated to cosmic-ray energies, we performed cosmicray transport calculations that account for the effects of propagation in the Galaxy and for solar modulation in the heliosphere using the models that we have discussed in Israel et al. (2018). Each of these processes is described by a set of linear differential equations: an operator  describing the propagation acts on a set of GCR source spectra, denoted by Q Z,A (E/A), to produce a corresponding set of equilibrium cosmic-ray energy spectra in the local ISM. Then, the operator  describing the solar modulation of the spectra as the cosmic rays penetrate to the inner heliosphere acts to produce the cosmic-ray energy spectra, denoted by j Z,A (E/A), that are observed near Earth. If Q represents the collection of all of the GCR source energy spectra of interest and  j represents the collection of all of the spectra observed near Earth, then we can write with  being the compound propagation/modulation operator describing the overall transport.
The individual operators  and  each depend on a variety of physical quantities. First,  depends on cross sections for fragmentation of cosmic-ray nuclei in collisions with atoms in the interstellar gas, as well as on half-lives for decays of radionuclides. Further, it depends on the diffusion coefficients describing how the spatial and energy distributions are affected by the transport, as well as on properties of the ISM and boundary conditions that affect the containment of cosmic rays. As is frequently done, we replace the complexities of these diffusion processes by a simple "escape length," Λ Z,A (E/A), giving the mean amount of interstellar material (measured in grams per centimeter squared) traversed by a nucleus of atomic number Z, mass number A, and energy per nucleon E/A before being lost from the system. We use the same form for Λ Z,A (E/A) that we employed in our earlier study of GCR elemental composition up to Z = 28 (Israel et al. 2018).
The solar modulation operator, , depends on the diffusion coefficient for transport in the interplanetary magnetic field, which in turn is a function of the small-scale turbulence in that field. To calculate the effects of modulation, we numerically solve a spherically symmetric Fokker-Planck equation (Goldstein et al. 1970;Fisk 1971) inside a free-escape boundary in the distant heliosphere. We neglect the effect of large-scale magnetic fields that can affect the flow pattern of cosmic rays through the heliosphere. Our modulation calculations use the same form of the interplanetary diffusion coefficient as in Israel et al. (2018), but with a modulation parameter f = 310 MV, which is taken to be an average value over the two solar cycles of our measurements. 8 To describe the energy spectra at the cosmic-ray source, we assume a common spectral shape ∝ (P/A) −2.35 for all nuclides, where P/A is the momentum per nucleon. This is consistent with the assumptions made in Israel et al. (2018). This spectral shape, multiplied by a nuclide's source abundance, q Z,A , gives the source spectrum used in our model, Because of the very limited statistics available for the measured ultraheavy galactic cosmic-ray (UHGCR) nuclides that are the subject of the present study, we do not have multiple observed spectral points. We combine all of the valid counts for each of the measured nuclides to derive average relative abundances (see Section 3) and treat those values as corresponding to the spectral intensities at an energy (E/A) 0 = 450 MeV/nuc. 9 Unlike our earlier work (Israel et al. 2018), where we derived source abundances by doing least-square fits to multiple measured spectral points, the single energy point available for the UHGCR nuclides allows us to use a simple matrix approach not requiring fitting. For each nuclide, we carry out the propagation/modulation calculation using a source abundance of unity for that nuclide and obtain the near-Earth abundances produced by that single-nuclide source. The values j j ((E/A) 0 ) obtained by propagating and modulating the source spectrum P A 2.35 ( ) -, corresponding to a unit abundance of nuclide i, make up one row in a matrix, M, expressing the linear relationship between a set of source abundances and the set of near-Earth abundances that result from the calculations. The inverse of this matrix multiplied into a vector of measured abundances gives the corresponding set of source abundances.
The source abundances obtained from this procedure are not guaranteed to all be nonnegative. Errors in the model parameters, particularly in cross sections for fragmentation production of secondary nuclides, can sometimes lead to negative values. Analysis of the CRIS data resulted in slightly negative source abundances for the nuclides 74 Se and 82 Kr. We set these source values to zero so that they will not lead to negative contributions when calculating secondary fractions in the abundances of nuclides arriving near Earth. More details of the transport model can be found in Appendix C.
In our source calculation, we initially derive isotope abundances relative to Cu, the most abundant element in the data set we are considering. In order to report values using a more conventional normalization, we have multiplied those results by 4.63 × 10 −4 , the Cu/Fe source-abundance ratio previously derived from ACE/CRIS observations (George et al. 2000). Table 2 summarizes the results of the source-abundance calculations. Column 3 lists the derived source abundances of each nuclide normalized to the element Fe. Columns 4 and 5 give the positive and negative uncertainties obtained from back-propagating the measured composition and uncertainties to the source. Column 6 shows the fraction of the measured abundance attributed to secondaries, as obtained from a forward propagation of the derived source composition. Estimates of the uncertainty in the derived source abundance of a nuclide due to uncertainties in the production cross sections used in the propagation were calculated assuming 20% completely correlated uncertainties in all of those cross sections (Israel et al. 2018), as described in Appendix C.2. Column 7 shows the resulting contributions to the source-abundance uncertainties. Since the cross-section uncertainties are poorly known, these can be taken as rough indicators at best. Our choice of 20% cross-section uncertainties is consistent with the approach taken by Israel et al. (2018) and with a comparison between calculated (Silberberg et al. 1998) and measured cross sections for 56 Fe fragmentation reported by Villagrasa-Canton et al. (2007). Israel et al. (2018) evaluated several other contributions to source-abundance uncertainties and found them to be small compared to the contribution associated with the production cross sections. Thus, we have neglected them in the present work. Columns 8 and 9 of the table show the sum in quadrature of the uncertainty associated with the measurements (columns 4 and 5) and with the transport calculations (column 7).

Discussion
We have presented here the first measurements of the isotopic abundances of elements of atomic number 31 Z 38 in GCRs. Also, we have presented abundances of the isotopes of Cu and Zn with greatly improved statistics over those previously reported (George et al. 1999(George et al. , 2000. We now compare the isotopic fractions of each of these elements in the GCRS with the corresponding fractions in the solar system (Lodders et al. 2009). Isotope fractions of a single element are unlikely to be significantly affected by effects at the source involving atomic properties like volatility, which has been shown to have a significant effect on the relative abundances of elements (Murphy et al. 2016;Lingenfelter 2019, and references therein). These GCRS isotopic fractions show generally good agreement with those in the solar system, as we show below.
The discussions by Murphy et al. (2016) and Lingenfelter (2019) treat the solar system abundances as representative of the general interstellar abundances that contribute to our observed cosmic rays, even though the solar system abundances reflect interstellar abundances ∼4.5 Gyr ago, while the observed cosmic rays were accelerated much more recently, ∼15 Myr ago; in the following discussion we make that same assumption. This assumption is supported by Prantzos (2008), in which he notes that "the elemental and isotopic abundances at solar birth [K] and today [K] are quite similar suggesting little chemical evolution in the past 4.5 Gyr." Further, Pinto et al. (2013) note that the ISM composition they measure today through X-ray spectroscopy "agree well with the abundances in the solar system as measured by Lodders et al. (2009)." In Table 3, "GCRS Fraction" (column 3) is the fraction of each element found in a particular isotope or isotope group in the GCRS, which was derived from Table 2. The "Statistical Uncertainty" (column 4) of the GCRS Fraction , where N is the total observed number of nuclei of this element (column 3 of Table 1).
We have assumed, as a worst case, that the "Propagation Uncertainty" (column 5) is dominated by uncertainty of the secondary production of this isotope during Galactic transport and that all these production cross-section uncertainties are completely correlated. This uncertainty depends on the secondary fraction of that isotope (column 7 of Table 2) and its derivation is described in Appendix C. The "Combined Uncertainty" (column 6) is the quadratic sum of the statistical and the propagation uncertainties. "SS Fraction" (column 7) is the corresponding isotopic fraction in the solar system (Lodders et al. 2009). "Delta" (column 8) is the difference between the GCRS Fraction and the SS Fraction. Column 9 gives Delta divided by the Combined Uncertainty of the GCRS Fraction.
We have omitted the element As from this table because it has only one stable isotope. We have also omitted Rb because CRIS recorded only seven Rb nuclei, without clear separation of isotopes.
The next four columns of Table 3 (columns 10 through 13) are taken directly from Table 2 of West & Heger (2013) and indicate the fraction of each isotope in the SS attributed to each of four nucleosynthesis sources: the Main s-process, the Weak s-process, the r-process, and Massive-star outflows. The "Massive" column of this table includes both production by  (  Table 3 of Prantzos et al. (2020). That paper did not analyze isotopes of Cu or Zn, so for those elements these rows are blank. It did not distinguish between the main s-process and the weak s-process; so for comparison with West & Heger (2013) column 14 is the sum of the two s-process columns of West & Heger (2013). Comparing columns 13 and 14 with columns 15 and 16, we note that these two papers agree on whether s-process or rprocess dominates the production of all the listed isotopes except 74 Ge, but they do not agree perfectly for the specific sand r-components.
We note that for most of the isotopes in Table 3 the magnitude of Delta is less than or about equal to the Combined Uncertainty, demonstrating that, to the precision of our GCRS results, for these isotopes the GCRS composition is consistent with that of the solar system. The two most notable exceptions to this agreement between the GCRS and the SS are 64 Zn and 67 Zn; however, since we do not know which nucleosynthesis process (or processes) contributes primarily to these isotopes, we do not attempt to draw any conclusions from these differences.
The most notable result that can be drawn from Table 3 is the lack of enrichment of the GCRS in isotopes that in the solar system are produced primarily in the r-process. For 76 Ge, 79 Br, and the heavy-isotope group of Kr, the isotopic fractions in the GCRS differ from those in the SS by less than half our GCRS uncertainty. For the heavy-isotope group of Se, the isotopic fraction in the GCRS is less than in the SS, but by less than twice the GCRS uncertainty.
This result is notable in light of previous observations regarding the GCRS. First, the elemental abundances of observed GCRs indicate that the GCR source is a mixture of ∼20% material produced by massive stars and ∼80% material with composition very similar to that of the solar system, presumed to be from the general Galactic ISM (Lingenfelter 2019, and references therein). Second, the observation of excess 22 Ne in the GCRS compared with the solar system (Binns et al. 2005, and references therein) is attributed to outflow of material from massive stars in their Wolf-Rayet stage, and thus supports a model where the GCRs are accelerated from the ISM in a region with greater density of massive stars than generally in the Galaxy-an OB association.
Core-collapse supernovae (CCSNe) are much more prevalent in OB associations than generally in the Galaxy, and 22 Ne in the GCRs demonstrates the enrichment of the ISM of OB associations in material synthesized by massive stars. So if interstellar r-process material were synthesized and ejected by CCSNe, one would expect the GCRS to be enriched in rprocess material relative to the solar system. Thus our observed lack of r-process enhancement in the GCRS is evidence that CCSNe are not the principal source of the lighter (Z 38) rprocess material found in the solar system.
We note that nucleosynthesis models as far back as the seminal paper by Burbidge et al. (1957) have pointed to supernovae as sites where high fluxes of neutrons should occur and thus might contribute to the production of r-process nuclides.
Our conclusion that CCSNe are not the principal source lends support to recent models in which most of the r-process material is synthesized in binary neutron-star mergers (Kasen et al. 2017;Sakari et al. 2018;Wanajo et al. 2021, and references therein).
We thank Nasser Barghouty for providing the program used for calculating production cross sections using the Silberberg et al. (1998) semiempirical formulas. We thank Lee Sobotka for providing advice and the program that enabled us to estimate the lifetime of those isotopes created during transport that decay primarily by electron capture, when the isotopes are fully stripped of their electrons. We thank Katharina Lodders for insightful discussion of the r-process in supernovae. And we thank Joe Giacalone for helpful comments about diffusive shock acceleration.
NASA grant Nos. NNX08AI11G, NNX13AH66G, and 80NSSC18K0223 supported the work at the California Institute of Technology, the Jet Propulsion Laboratory, and Washington University in St. Louis. The work at Goddard Space Flight Center was funded by NASA through the ACE project.

Appendix A Corrections to Measurements
As discussed in Section 4, we have made three corrections to the measured numbers of counts to obtain isotope abundances at the top of the instrument: (1) for saturation in the PHAs; (2) for nuclear interactions occurring inside the instrument; and (3) for the differing energy intervals of each isotope. This appendix gives further details about those corrections.

A.1. Saturation Corrections
The CRIS PHAs were designed for an instrument that would only study nuclei of atomic number (charge) Z 30. In this paper we are presenting results for elements with 29 Z 38. In the interval 32 Z 38, for some particles the PHAs for one or more of the detectors saturated, depending on the particle's charge, mass, angle of incidence, and its energy at each detector. Particles for which any of the detector pulse heights were saturated were excluded from analysis, resulting in a decrease of detection efficiency for affected isotopes. A Monte Carlo calculation was performed to determine the fraction of particles as a function of charge (Z) and atomic mass (A) that saturated in one or more detectors. Saturation differences between different detector stacks were small, so average values were used. This calculation was done separately for particles stopping in each range (E2 through E8), using the measured thickness of each silicon detector and applying the same restrictions on stopping depth that are used in the analysis of the data. The Monte Carlo calculation was done individually for each stable isotope of each element with Z 30 using large enough numbers of input particles to make the statistical uncertainties in the derived saturation corrections negligible. (For Z < 32, saturated signals did not occur in any of the detectors for any of the particles.) Trajectories simulated of particles incident on CRIS were isotropic over the allowed angles of incidence (0°-45°). Table 4, column 3 shows the saturationcorrection factors obtained in the Monte Carlo calculation.

A.2. Nuclear-interaction Corrections
As described in Sections 2 and 4, above, we rejected events where an incident nucleus suffered a nuclear interaction within the instrument. To make interaction corrections to the measured isotope abundances, we assume that we have successfully rejected all interacting particles by our selections and that the number of interactions for each isotope can be correctly estimated using the Westfall et al. (1979) mass-changing cross sections. Most mass-and charge-changing interactions are rejected by our selection criteria, since either a change in mass or a change in charge of the incident nucleus should result in an inconsistency in the multiple mass estimates made by CRIS. The exception is particles stopping in range 2 (see Section 2, above), since only a single mass estimate is available for those nuclei. Corrections are required for both charge-changing and mass-changing interactions. The Westfall et al. (1979) masschanging cross sections incorporate the effects of both chargeand mass-changing interactions. 10 To estimate the number of events so rejected, the Westfall et al. (1979) mass-changing total interaction cross-sections were utilized. For each range the mean of the minimum and maximum amount of silicon-equivalent material traversed (particles with incidence angle normal to the detectors and with incidence angle 45°, respectively) was calculated. The fraction of events lost to interactions for each mass was then calculated for each range. These fractions were then averaged, weighted by the number of iron events stopping in each range, to estimate the fraction lost to interactions for each isotope. Table 4, column 4, shows the corrections relative to 56 Fe that were obtained (the 56 Fe/Fe ratio equals 0.81).
Of the events rejected as nuclear interactions in the instrument because of disagreement among the multiple charge determinations, a small fraction may have been the result of channeling (Gemmell 1974;Nordlund et al. 2016, and references therein) in one or more of the silicon detectors rather than nuclear interactions. We have not tried to correct for such events because their occurrence probabilities would be similar for all isotopes of a given element and so would not significantly affect the isotope fractions.

A.3. Energy-interval Corrections
For each Z and A, the energy interval analyzed extended from the lowest energy for which a normal-incidence (0°) nucleus could be analyzed in E2, to the highest energy for which a nucleus incident at 45°could stop in the useful part of E8. Thus different nuclides had energy intervals of different widths over different parts of the energy spectrum. The minimum and maximum areal densities from the top of the instrument were calculated, with the minimum corresponding to a particle having its trajectory normal to the detector surfaces stopping 25 μm into E2 and the maximum corresponding to a 45°particle stopping at a normal depth of 5850 μm in E8. The minimum and maximum energies for each isotope were then calculated based on range-energy and stoppingpower tables for protons (Berger et al. 2017). 11 The proton ranges (R p ) were then scaled using the relationship R(A, Z, E/A) = (A/Z 2 ) × R p (E/A) to obtain the range of the isotope of interest. The integral flux was then calculated for each isotope over its energy interval, assuming that all isotopes have the same spectral shape as Fe. An Fe energy spectrum (see Israel et al. 2018, Figure 2) was derived from the CRIS Level 2 data 12 and is shown in Figure 5. This is shown as a semi-log plot, rather than a log-log plot, to better illustrate the difference in energy intervals covered by each isotope. Energy intervals for three example isotopes, 56 Fe, 74 Ge, and 88 Sr, are shown in the figure. The integral flux was then calculated for the energy interval corresponding to each isotope by integrating the Fe energy spectrum over its energy interval. The reciprocal of the ratio of each calculated flux relative to that for 56 Fe then is the energy-interval correction factor for that isotope ( Table 4, column 5).

A.4. Combined Corrections
The product of the three correction factors then gives the overall correction factor relative to 56 Fe for each isotope in Table 4, column 6, and relative to the most abundant isotope for that element in column 7. We note that for Kr and Sr the corrections for PHA saturation relative to 56 Fe dominate and are large (columns 3 and 6). However, for estimating the isotope fractions of each element relative to its most abundant isotope ( Table 4, column 6), statistical uncertainties dominate.
The combined correction factors relative to 56 Fe (column 6) are those plotted in the upper panel of Figure 6. The lower panel of Figure 6 shows the contribution of each of the three correction factors for each element and mass.

A.5. Correction Factors for Isotope Groups
To calculate the correction factors for isotope groups for Z = 34-38, weighted averages of the individual isotope correction factors were used, with weighting done using the measured fractional abundances of the individual isotopes. The weights are listed in column 8 of Table 4. Column 9 shows the resulting weighted average correction factors. The element abundances relative to Fe were determined in two ways. The first was to take cuts between the isotope groups in the pseudo-Z histogram (Figure 2). The second was simply to sum the isotopes for each element listed in Table 1.
The differences between these methods of obtaining abundances were small, so we took the average of the two estimates as representing our best estimate of element abundances. Table 5 presents the element abundances. Columns 2 and 3 are the counts for each element obtained by cutting the pseudo-Z histogram and by summing the isotopes for each element, and column 4 is the mean of these two methods of counting. Columns 5 and 6 are statistical uncertainties assuming Poisson statistics (Gehrels 1986). Column 7 gives the estimated systematic uncertainties. For Cu these were calculated by taking half of the difference between the two methods of counting particles. Since this is the largest systematic difference for any of the elements, and since the systematic errors become much smaller than the statistical errors for higher charges, we use the fractional systematic uncertainty for Cu, which is 0.0223, for all higher-Z nuclei. Columns 8 and 9 give the quadratic sum of the statistical and systematic errors, and we take this to be the overall uncertainty for the elements. Column 10 gives the combined correction factors for elements. This was calculated by taking the average of the isotope correction factors for a given element, weighted by the number of counts found for each isotope using the GML fit. Columns 11 through 13 give the corrected elemental abundances and uncertainties relative to Fe = 10 6 . In Figure 7 we compare our CRIS elemental abundances with values from the balloon-borne SuperTIGER experiment reported by Murphy et al. (2016) and Walsh (2020). For Z 36 we see that the CRIS abundances tend to be somewhat lower than those of SuperTIGER. We do not know the reason for this, but a possibility is that it results from the different energy ranges measured by CRIS and SuperTIGER, which measures particles with energy 1 GeV/nuc.    Corrected Uncert (−) rel to Fe = 10 6 (1) ( Appendix C Propagation Model

C.1. Leaky-box Model
For calculating the effects of interstellar propagation, we use the simple leaky-box model that has been described by Israel et al. (2018) and is based on Meneguzzi et al. (1971). For each nuclide, i, we denote the spectrum of accelerated particles at the cosmic-ray source by q i (ò), where ò is the energy per nucleon. The model accounts for losses of accelerated particles due to a variety of mechanisms including nuclear fragmentation reactions and, in the case of radioactive nuclides, radioactive decay. In addition, particles undergo diffusive transport in the Galaxy as well as transport in energy space due to ionization energy loss that occurs in traversing the ISM. The leaky-box model uses a highly simplified approximation, replacing the spatial diffusion with an empirical "escape mean free path" representing the mean amount of interstellar matter (grams per centimeter squared) traversed (Jones et al. 2001). This mean free path, denoted by i esc  ( ) L , depends on the nuclide being considered and its energy per nucleon. In this study, we obtained i esc  ( ) L from Israel et al. (2018). Given this set of physical processes, the relationship between the source spectrum and the resulting equilibrium interstellar spectrum, j i (ò), is given by a continuity equation describing the balance between gains and losses of particles of each nuclide at each energy.
It is convenient to distinguish between "primary" particles, , accelerated at the cosmic-ray source and "secondary" particles,  , produced by mechanisms such as nuclear fragmentation and radioactive decay during propagation in the Galaxy. Thus we write are the mean free paths for loss due to escape, fragmentation, and radioactive decay, respectively, of nuclide i. Each of these can also be a function of energy. The final term on the right describes energy changes due to ionization energy loss in the ISM, with w i (ò) representing the specific ionization per nucleon.
As discussed in Section 5, we assume that the primaries are produced with a source spectral shape in the form of a power law in momentum per nucleon. Although diffusive shock acceleration theory (e.g., Blandford & Ostriker 1978;Jones & Ellison 1991) predicts a momentum power law as the spectral shape for any particular nuclide, the theory taken alone does not address questions of the relative efficiency of acceleration of different nuclides. Our use of the source spectrum, with the subscript i denoting the nuclide with atomic number Z i and mass number A i , implicitly makes an assumption about these relative efficiencies that is relevant when we compare the resulting source abundances, q i , with the composition of matter in the solar system. Although the theory does not provide a justification for that assumption, the generally good agreement that we obtain between GCRS and solar system abundances of refractory elements and isotopes (Wiedenbeck et al. 2007;Israel et al. 2018, and references therein) suggests that it is not significantly in error.
Secondary production by fragmentation of heavier nuclides has the form (Meneguzzi et al. 1971) Here ji * s is the weighted average cross section for production of nuclide i from fragmentation of nuclide j in collisions with interstellar H and He in a number-density ratio n He /n H = 0.1, and M * is the weighted average mass of these target atoms. For nuclides that can be produced by radioactive decay, an additional term (not shown) is included in i  to account for that production and the effect of these decays is included in the loss mean free path .
i decay L Equation (C1) is the "leaky-box equation" that is solved to obtain equilibrium interstellar spectra from the source spectra. In cases where energy loss effects are small enough to be neglected, the rightmost term can be omitted, leaving an algebraic equation. The energies at which we measure nuclides in the iron group and above (see Figure 5) combined with the mean energy loss experienced due to solar modulation in penetrating to Earth (∼300 MeV/nucleon) place the energies of interest near the "minimum ionization" point on the curve of w i (ò) versus ò. We do include the energy losses in our propagation calculation, but they play only a minor role.

C.2. Propagation Uncertainties
To calculate the uncertainty in the derived source abundance of a particular nuclide due to measurement uncertainties (statistical and systematic added in quadrature), we simply repeat the propagation with each of the near-Earth abundances adjusted in turn by ±1 standard deviation. The calculation includes the effects of both propagation and solar modulation. These uncertainties are listed in columns 4 and 5 of Table 2.
Uncertainties in the production of secondary cosmic rays affect the derived source abundances because the secondaries must be subtracted from the observed abundances to derive the amount of surviving primary material, which gets used to obtain the source abundance. Thus, the ratio between the absolute uncertainty in i  and the uncertainty it produces in the source abundance is the same as the ratio between the measurement uncertainty in i  and the uncertainty it produces in the source abundance.
We make the conservative assumptions that all of the secondary production cross sections have a fractional uncertainty f and that the uncertainties in all of these cross sections are completely correlated. (The case of completely uncorrelated cross-section uncertainties, which would lead to smaller source-abundance uncertainties, has been discussed by Israel et al. 2018, where it was found that the cross-section-related uncertainties were smaller in the completely uncorrelated case than in the completely correlated case by factors that ranged between 0.24 and 0.80.) In this case, i  has this same fractional uncertainty, f, and the corresponding absolute uncertainty is f times the secondary fraction, that is equal to f times the nuclide's secondary fraction in the observed cosmic rays. The secondary fraction is listed in column 6 of Table 2 and the resulting "± Error from Propagation" in the source abundance is in column 7. Finally, we add the values in columns 4 and 7 in (16)