Isotopic dependence of GCR fluence behind shielding

Radiation Measurements 41 (2006) 1235–1249www.elsevier.com/locate/radmeas

Isotopic dependence of GCR fluence behind shielding

Francis A. Cucinottaa,∗, John W. Wilsonb, Premkumar Sagantic, Xiaodong Hua,Myung-Hee Y. Kima, Timothy Cleghorna, Cary Zeitlind, Ram K. Tripathib

aNASA, Lyndon B. Johnson Space Center, Houston, TX 77058, USAbNASA, Langley Research Center, Hampton, VA 23664, USA

cPrairie View A&M, Prairie View, TX 94720, USAdLawrence Berkeley National Laboratory Berkeley, CA 94720, USA

Received 26 October 2005; received in revised form 8 December 2005; accepted 8 March 2006

Abstract

In this paper we consider the effects of the isotopic composition of the primary galactic cosmic rays (GCR), nuclear fragmentation crosssections, and isotopic-grid on the solution to transport models used for shielding studies. Satellite measurements are used to describe theisotopic composition of the GCR. For the nuclear interaction data-base and transport solution, we use the quantum multiple scattering theoryof nuclear fragmentation (QMSFRG) and high-charge and energy (HZETRN) transport code, respectively. The QMSFRG model is shown toaccurately describe existing fragmentation data including proper description of the odd–even effects as function of the iso-spin dependence onthe projectile nucleus. The principle finding of this study is that large errors (±100%) will occur in the mass-fluence spectra when comparingtransport models that use a complete isotopic-grid (∼ 170 ions) to ones that use a reduced isotopic-grid, for example the 59 ion-grid used inthe HZETRN code in the past; however, less significant errors (<± 20%) occur in the elemental-fluence spectra. Because a complete isotopic-grid is readily handled on small computer workstations and is needed for several applications studying GCR propagation and scattering, it isrecommended that they be used for future GCR studies.© 2006 Elsevier Ltd. All rights reserved.

Keywords: Space radiation shielding; Nuclear fragmentation; Galactic cosmic rays

1. Introduction

An important goal for NASA’s Space Radiation Health Pro-gram is to develop a predictive capability to predict the GCRfluence spectra to within a±25% accuracy (Anonymous, 1998).NASA has developed the HZETRN (high-charge and energytransport) code (Wilson, 1977; Wilson and Badavi, 1986; Wil-son et al., 1991) as a science application and engineering designtool (Wilson et al., 1993) to be used in space radiation shield-ing studies. HZETRN has been validated in its ability to pre-dict total dose and dose equivalent behind several materials inspace to within±20% on multiple space missions in Earth orbit(Cucinotta et al., 2000; Badhwar and Cucinotta, 2000; Badhwaret al., 2001). However, interest in fluence-based approaches to

∗ Corresponding author. Tel.: +1 281 483 0968; fax: +1 291 483 3058.E-mail address: [email protected] (F.A. Cucinotta).

1350-4487/$ - see front matter © 2006 Elsevier Ltd. All rights reserved.doi:10.1016/j.radmeas.2006.03.012

risk assessment (Cucinotta and Wilson, 1995; Cucinotta et al.,1996) suggests that more stringent tests of transport code ac-curacy be made, and the quantities dose and dose equivalentare deemed as necessary, but not sufficient tests of their accu-racy. In this regard, we note that dose and dose equivalent areintegral quantities that receive contributions from many GCRcharge groups. There currently exist large uncertainties in bio-logical response models for GCR (Cucinotta et al., 2001) andmethodologies to estimate health risks such that dose and doseequivalent may be insufficient as tests of transport code accu-racy. The use of ion fluence as a basis for tests for accuracyprovides for sufficient generality to ensure accuracy in GCRtransport models including under the circumstances of revisionof radiation quality factors or integration of alternative risk as-sessment approaches in the future.

In the description of the transport of the galactic cos-mic rays (GCR) in shielding materials or tissue, a common

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1236 F.A. Cucinotta et al. / Radiation Measurements 41 (2006) 1235–1249

approximation is to consider only the elemental compositionof the primary GCR and a reduced isotopic-grid for the sec-ondary nuclei produced in nuclear fragmentation. In this paperwe analyze the role of the isotopic dependence of the GCRprimary composition and nuclear fragmentation in predictingthe fluence of the GCR behind arbitrary shielding configura-tions. Our study is an important milestone in achieving NASA’sgoal of accurate GCR transport codes, since for the first-timea complete isotopic-grid has been achieved in a GCR trans-port model and we document the error inherent in former ap-proaches. Also, for applications that will consider radioactiveisotopes produced in the atmosphere or shielding, our studyprovides a useful tool to perform such analyses. Other applica-tions where non-stable nuclei are considered are studies of theorigin or the GCR where so-called cosmic-ray “clocks’’ con-sider the primary or secondary GCR with life-times on the or-der of confinement time in the galaxy (∼ 1 M yrs) (Yanasaket al., 1999). Several GCR “clock’’ nuclei including 10Be and26Al were not considered in the grid used in HZETRN in thepast. Finally, new data on the GCR near Mars are being col-lected by the MARIE experiment on the Odyssey spacecraft(Zeitlin et al., 2004), and our study provides an opportunityto begin new investigations on the accuracy of computationalmodels used to describe the GCR.

Historically the HZETRN code grew from a 29-ion isotopicgrid used in the 1980s and early 1990s (Wilson et al., 1991)to an extension to a 32-ion isotopic grid made in 1993 in or-der to include all light ions (Cucinotta, 1993). Because of thelimitations of random access memory present on the computerworkstations of the early 1990s, sensitivity studies were madefor mono-energetic ion beams to study the minimum numberof isotopes for convergence resulting in the use of 59-isotopicgrid (Kim et al., 1994), and all GCR studies since 1994 haveused the 59-isotopic grid (Shinn et al., 1994). However, thereare several reasons to re-consider the use of the full isotopic-grid for GCR transport problems. First, the isotopic depen-dence of the primary GCR has not been considered in pastshielding studies and may lead to errors in the descriptionof both primary ion attenuation and secondary particle pro-duction including the role of high-energy neutron productionfrom the many neutron rich species that occur. Secondly, thestudies of Kim et al. (1994) used the NUCFRG2 model offragmentation (Wilson et al., 1994), which does not providea correct description of the even–odd effect observed in frag-ment production or of the projectile iso-spin dependence ob-served experimentally (Knott et al., 1997; Zeitlin et al., 2001).Thirdly, the sensitivity studies made by Kim et al. (1994) useda “calibration’’ of the isotopic-grid to 56Fe beams; however, alarger isotopic grid occurs when all GCR projectile nuclei areconsidered. Fourth, the error in the range energy and stoppingpowers that results from the use of a reduced isotopic grid, al-though expected to be small for large mass number, A?1, isan unnecessary one for transport calculations. Finally, the im-proved computational speed and memory available on currentsmall computer workstations, makes the inclusion of a completeisotopic-grid in the HZETRN code to be readily implemented atthis time.

In this paper the implementation of the HZETRN code toinclude the full isotopic dependence of the primary GCR is de-scribed. The physics of isotopic effects in GCR transport aredescribed and the fragmentation parameters are a key compo-nent of this description. The quantum multiple scattering theoryof nuclear fragmentation (QMSFRG) is used as the generatorfor fragmentation cross sections used in our study. An empiri-cal model of the isotopic composition of the primary GCR in-cluding its solar modulation is also described. For GCR prob-lems an isotopic grid of 170 ions is identified and comparisonsmade to previous HZETRN results using the reduced-grid of 59isotopes are made. The present code includes all of the abun-dant nuclei in the GCR environment with fluxes greater thanabout 102/cm2/yr and nuclei produced in fragmentation eventswith production cross sections greater than about 1 mb. Severalnuclei with smaller primary abundances or production crosssections, which are of interest for scientific reasons are alsoincluded in the expanded HZETRN model described herein.The resultant code includes many neutron rich nuclei that havebeen ignored in the past with iso-spin components ranging fromTz =+3/2 to Tz =−3.

2. Isotopic composition of the GCR

NASA currently uses the GCR model of Badhwar andO’Neill (1992) to describe the elemental composition andenergy spectra of the GCR including their modulation by thesun’s magnetic field. In this GCR representation, only the mostabundant GCR nuclei is considered for each element and otherisotopes of identical charge are counted as the abundant iso-tope. However, theoretical models and satellite measurementsof the GCR have long considered the isotopic composition ofthe GCR and their modification through transport in interstellarspace including estimating the primary nuclear compositionat stellar sources (Parker, 1965; Webber et al., 1990a; Fieldset al., 1994). The approach used here is to estimate an energy-independent isotopic fraction, fj from satellite measurements,which are constrained to obey the sum rule

�(Z, E)=∑Aj

fj (Aj , Z)�(Aj , Z, E), (1)

where the left-hand side of Eq. (1) is the elemental spectrafrom the Badhwar and O’Neill model and

∑j fj = 1. Eq. (1)

is used herein as an initial estimate of the influence of the pri-mary isotopic composition on GCR shielding calculations. Ex-perimental studies have included measurements on the Pioneer,Voyager, and Ulysses spacecraft. A survey of such data (Hesseet al., 1991; Lukasiak et al., 1993, 1995; Webber et al., 1985,1990a; Wiedenback and Greiner, 1981; Wiedenback, 1985) wasmade with the results shown in Table 1.

The GCR path-length distribution represents the meanamount of inter-stellar and inter-planetary material intersectedby cosmic rays prior to their arrival in the near-Earth environ-ment. This distribution is currently estimated to vary between3 and 20 g/cm2 with the inter-planetary gas approximately90% H and 10% He. Variability in estimates between various

F.A. Cucinotta et al. / Radiation Measurements 41 (2006) 1235–1249 1237

Table 1Isotopic composition of GCR elements: (a) Z= 3–12, (b) Z= 13–20 and (c)Z = 22–26

Isotope Near-Earth fraction Source fraction

(a)Z = 3

6Li* 0.5 0.57Li 0.5 0.5

Z = 47Be* 0.5 0.59Be 0.35 0.3510Be 0.15 0.15

Z = 510B 0.31 0.211B 0.69 0.8

Z = 612C 0.92 0.99913C 0.08 0.001

Z = 714N 0.48 0.7815N 0.57 0.22

Z = 816O 0.946 0.98517O 0.027 0.00818O 0.027 0.007

Z = 1020Ne 0.55 0.6821Ne 0.10 0.022Ne 0.35 0.32

Z = 1224Mg 0.64 0.7425Mg 0.18 0.1426Mg 0.18 0.13

(b)Z = 13

26Al 0.02 0.027Al 0.98 1.0

Z = 1428Si 0.84 0.90229Si 0.08 0.05430Si 0.08 0.044

Z = 1632S 0.69 0.9633S 0.15 0.0234S 0.16 0.02

Z = 1735Cl 0.52 1.036Cl 0.41 0.037Cl 0.26 0.0

Z = 1836Ar 0.64 1.037Ar 0.03 0.038Ar 0.30 0.040Ar 0.03 0.0

Z = 2040Ca 0.4 1.041Ca 0.2 0.042Ca 0.2 0.043Ca 0.2 0.044Ca 0.2 0.0

Table 1 (continued)

Isotope Near-Earth fraction Source fraction

(c)Z = 22

46Ti* 0.27 0.2747Ti 0.31 0.3148Ti 0.30 0.3049Ti 0.09 0.0950Ti 0.03 0.03

Z = 2349V* 0.53 0.5350V 0.24 0.2451V 0.23 0.23

Z = 2450Cr* 0.26 0.2651Cr 0.26 0.2652Cr 0.48 0.48

Z = 2553Mn* 0.43 0.4354Mn 0.17 0.1755Mn 0.40 0.40

Z = 2654Fe 0.076 0.05555Fe 0.084 0.07856Fe 0.763 0.79257Fe 0.076 0.075

∗Data on solar modulation was not found and thus near-Earth and sourcecomposition are set equal.

studies arise because of differences in model nuclear fragmen-tation parameters, cosmological interaction terms, and factorsrelated to the near-Earth or deep space measurements includ-ing energy and mass resolution of the detectors as well asposition in the solar cycle. In most current models, the GCRpath-length distribution is estimated to be velocity and rigiditydependent (i.e., energy dependent). Of note is that based on thesolar modulation theory of Parker (1965), the inter-planetaryportion of this distribution would be modulated over the solarcycle suggesting the form

�(�, R)= �Inter-stellar + �Inter-planetary(�, R), (2)

where � is the solar modulation parameter and R (in A.U.) isthe radial distance from the sun. It then follows that the amountof fragmentation in the inter-planetary media and the isotopicabundances are dependent on the modulation parameter andradial distance. Based on the results of Lukasiak et al. (1993)the following formula is used to describe the dependence ofthe isotopic ratio (near-Earth) on the solar cycle including adependence on the modulation parameter �(MV)

f (A, Z)= fSource(A, Z)+ [exp(�√

�)− 1], (3)

where � is found by interpolating from the source and near solarmaximum values. The second term in Eq. (3) can be positiveor negative as dependent on the conditions for a net gain or


0 400 800 1000

Isot

opic

Rat

io (

%)

0

20

40

60

13C15N

Modulation Parameter, Φ (MV)

600200

Fig. 1. Parametric model for describing the change in isotopic compositionwith the solar modulation parameter, �(MV).

reduction in the cosmic nuclei during transit from the galacticsources (Fields et al., 1994) to the inner heliosphere. In Table 1we have used data on isotopic fractions near solar maximumwhere the isotopic fraction for nuclei produced within the he-liosphere are expected to be at a maximum (Hesse et al., 1991;Lukasiak et al., 1993; Webber et al., 1985, 1990a; Wiedenbackand Greiner, 1981; Wiedenback, 1985). For several elementslisted in Table 1, information on solar modulation was not avail-able and isotopic fractions for near-Earth and GCR source wereset identical using the near-Earth estimate.

We represent the primary (near-Earth) 2H and 3He spectra as

�3He(E)= {0.0764+ 0.097 exp

× [−0.5(ln(E/1660)/1.306)2]}�4He(E), (4a)

�2H(E)= 0.2�4He(E), (4b)

where �3He(E) and �2H(E) are subtracted from the Badhwarand O’Neill model (1992) for �4He(E) and �1H(E), respec-tively. Fig. 1 shows the assumed dependence of the isotopeabundance scaling as a function on the solar modulation pa-rameter described by Eq. (3) for 13C and 15N. Similar resultsare found for other primary isotopes listed in Table 2. Exam-ples of the GCR energy spectra for hydrogen and helium areshown in Fig. 2a and for the isotopes of Ne, Si, and Fe at solarminimum in Fig. 2b.

3. Isotopic effects in GCR transport

The reduction of the full three-dimensional Boltzmann trans-port equation within the straight-ahead and continuous slow-ing down approximations and assuming velocity conservativefragmentation events accurately describes the transport of GCRheavy ions (Wilson and Badavi, 1986; Wilson et al., 1991,1995). In this model the heavy ion flux, �j (E, x) of an ion jwith mass number Aj , charge number Zj, energy E (in units

of MeV/u) at shielding depth x (in units of g/cm2) is determinedby the partial differential equation (Wilson et al., 1991)[

�

�x− 1

Sj (E)

�

�E+ �j (E)

]�j (x, E)

=∑

k

�j,k(E)�k(E, x), (5)

where �j (E) is the energy-dependent absorption cross section(cm−1) and �jk(E) is the fragmentation cross section for pro-ducing an ion j from k. The solution to Eq. (5) in the HZETRNcode is found using the methods of characteristics where the co-ordinate transformation from depth and particle range, Rj (E),are defined

�j = x − Rj (E), �j = x + Rj (E) (6a)

and the scaled flux

j (�j , �j )= Sj (E)�j (x, E) (6b)

are introduced leading to the transport equation[2

�

��j

+ �j

](�j , �j )=

∑k

�j,k

j

k

(�k, �k), (7)

where Sj (E) is the stopping power of ion j, vj = Z2j /Aj , �j

the total absorption cross section, and �jk the fragmentationcross section to produce isotope j from projectile isotope k.The numerical solution to Eq. (7) uses a marching procedure(Wilson et al., 1995).

The solution of the transport equation for light ions andneutrons is distinct from Eq. (7) because of the broad re-distribution of energy in collision events (Wilson et al., 1991;Cucinotta et al., 1995; Clowdsley et al., 2001). A coupledintegro-differential equation has been formulated for neutrons,and hydrogen and helium isotopes. As described previously amarching procedure is introduced, which can be solved exactlyto second order in the step-size, h which is chosen as 0.5 g/cm2

after convergence tests (Wilson et al., 1995). An important dis-tinction from the heavy ion part of the transport code, is that forlight ions models of the differential in energy production crosssections for knockout, cascade, and evaporation are needed,and for neutrons, elastic scattering must be treated.

Because Eq. (7) is a coupled differential equation for themany GCR primary and secondary nuclei, required computermemory allocations increase rapidly as the number of ionspecies is increased and was an important consideration in theearly 90’s. However, such practical limitations no longer exist,even on small computer workstations with sufficient RAM(e.g. Pentium III or higher with 0.5 Giga-bytes or more), andan unrestricted isotopic-grid can presently be implementedwith no memory or storage problems. When using a reducedisotopic grid, fragments not contained in the grid are assignedto a nearby mass of identical charge number. This introducestwo types of errors: first the range-energy and stopping powersare altered from their true values. This error is expected to besmall at high-energies for A?1 since here ion ranges are welldescribed by Aj/Z

2j scaling factors. The second type of error


Table 2Isotopic grid used in present HZETRN calculations for elements: (a) Z = 0–11, (b) Z = 12–18 and (c) Z = 19–28

Index, j Nuclei Tz Lifetime Decay Index, j Nuclei Tz Lifetime Decay

(a)1 1n −1/2 614.6 s �− 24 12N +1 0.0111 s e+,�+2 1H +1/2 Stable 27 13N +1/2 9.965 m e+,�+3 2H 0 Stable 29 14N 0 Stable4 3H −1/2 12.33 y �− 32 15N −1/2 Stable5 3He +1/2 Stable 34 16N −1 7.13 s �−6 4He 0 Stable 36 17N −3/2 4.173 s �−7 6He −1 0.807 s �− 30 14O +1 70.606 s e+,�+8 6Li 0 Stable 33 15O +1/2 122.24 s e+,�+9 7Li −1/2 Stable 35 16O 0 Stable

11 8Li −1 0.838 s �− 37 17O −1/2 Stable13 9Li −3/2 0.178 s �− 39 18O −1 Stable10 7Be +1/2 53.12 d e 42 19O −3/2 26.91 s �−14 9Be −1/2 Stable 45 20O −2 13.51 s �−16 10Be −1 1.51× 106 y �− 38 17F +1/2 64.49 s e+,�+19 11Be −3/2 13.81 s �− 40 18F 0 0.1098 s e+,�+12 8B +1 0.770 s �+ 43 19F −1/2 Stable15 9B +1/2 0.54 keV 2�p 46 20F −1 11.0 s �−17 10B 0 Stable 48 21F −3/2 4.158 s �−20 11B −1/2 Stable 41 18Ne +1 1.672 s e+,�+22 12B −1 0.0202 s �− 44 19Ne +1/2 17.22 s e+,�+25 13B −3/2 0.01736 s �− 47 20Ne 0 Stable18 10C +1 19.255 s e+,�+ 49 21Ne −1/2 Stable21 11C +1/2 0.0204 s e+,�+ 51 22Ne −1 Stable23 12C 0 Stable 53 23Ne −3/2 37.24 s �−26 13C −1/2 Stable 56 24Ne −2 3.38 m �−28 14C −1 5730 y �− 50 21Na +1/2 22.49 s e+,�+31 15C −3/2 2.449 s �− 52 22Na 0 2.6019 y e+,�+

54 23Na −1/2 Stable57 24Na −1 14.959 h �−59 25Na −3/2 59.1 s �−62 26Na −2 1.072 s �−

(b)55 23Mg +1/2 11.317 s e+,�+ 78 30S +1 1.178 s e+,�+58 24Mg 0 Stable 81 31S +1/2 2.572 s e+,�+60 25Mg −1/2 Stable 84 32S 0 Stable63 26Mg −1 Stable 87 33S −1/2 Stable66 27Mg −3/2 9.458 m �− 90 34S −1 Stable69 28Mg −2 20.91 h �− 93 35S −3/2 87.32 d �−61 25Al +1/2 7.183 s e+,�+ 96 36S −2 Stable64 26Al 0 7.17× 105 y e+,�+ 99 37S −5/2 5.05 m �−67 27Al −1/2 Stable 103 38S −2 170.3 m �−70 28Al −1 2.241 m �− 88 33Cl +1/2 2.511 s e+,�+72 29Al −3/2 6.56 m �− 91 34Cl 0 1.5264 s e+,�+75 30Al −2 3.60 s �− 94 35Cl −1/2 Stable65 26Si +1 2.234 s e+,�+ 97 36Cl −1 3.01× 105 y �−68 27Si +1/2 4.16 s e+,�+ 100 37Cl −3/2 Stable71 28Si 0 Stable 104 38Cl −2 37.34 m �−73 29Si −1/2 Stable 108 39Cl −5/2 55.60 m �−76 30Si −1 Stable 95 35Ar +1/2 1.775 s e+,�+79 31Si −3/2 0.1573 s �− 98 36Ar 0 Stable82 32Si −2 150 y �− 101 37Ar −1/2 35.04 d e85 33Si −5/2 6.18 s �− 105 38Ar −1 Stable74 29P +1/2 4.140 s e+,�+ 109 39Ar −3/2 269 y �−77 30P 0 2.498 m e+,�+ 112 40Ar −2 Stable80 31P −1/2 Stable 115 41Ar −5/2 109.34 m �−83 32P −1 14.262 d �− 118 42Ar −3 32.9 y �−86 33P −3/2 25.34 d �−89 34P −2 12.43 s �−92 35P −5/2 47.3 s �−


Table 2 (continued)

Index, j Nuclei Tz Lifetime Decay Index, j Nuclei Tz Lifetime Decay

(c)102 37K +1/2 1.226 s e+,�+ 134 46V 0 0.422 s e+,�+106 38K 0 7.636 m e+,�+ 137 47V −1/2 32.6 m e+,�+110 39K −1/2 Stable 140 48V −1 15.9735 d e+,�+113 40K −1 1.277× 109 y �− 143 49V −3/2 330 d e116 41K −3/2 Stable 146 50V −2 1.4× 1017 y e+,�+119 42K −2 12.36 h �− 149 51V −5/2 Stable122 43K −5/2 22.3 h �− 152 52V −3 3.743 m �−107 38Ca +1 0.440 s e+,�+ 141 48Cr 0 21.56 h e+,�+111 39Ca +1/2 0.8596 s e+,�+ 144 49Cr −1/2 42.3 m e+,�+114 40Ca 0 Stable 147 50Cr −1 > 1.8× 1017 y e117 41Ca −1/2 1.03× 105 y e 150 51Cr −3/2 27.7025 d e120 42Ca −1 Stable 153 52Cr −2 Stable123 43Ca −3/2 Stable 155 53Cr −5/2 Stable125 44Ca −2 Stable 158 54Cr −3 Stable128 45Ca −5/2 162.61 d �− 148 50Mn 0 0.284 s e+,�+131 46Ca −3 Stable 151 51Mn −1/2 0.0462 s e+,�+121 42Sc 0 0.681 s e+,�+ 154 52Mn −1 5.591 d e+,�+124 43Sc −1/2 3.891 h e+,�+ 156 53Mn −3/2 3.74× 106 e126 44Sc −1 3.927 h e+,�+ 159 54Mn −2 312.3 d e+,�+129 45Sc −3/2 Stable 161 55Mn −5/2 Stable132 46Sc −2 83.79 d �− 157 53Fe −1/2 0.00851 s e+,�+135 47Sc −5/2 3.3492 d �− 160 54Fe −1 Stable138 48Sc −3 43.67 h �− 162 55Fe −3/2 2.73 y e127 44Ti 0 63 y e 164 56Fe −2 Stable130 45Ti −1/2 184.8 m e+,�+ 167 57Fe −5/2 Stable133 46Ti −1 Stable 163 55Co −1/2 17.53 h e+,�+136 47Ti −3/2 Stable 165 56Co −1 77.27 d e+,�+139 48Ti −2 Stable 168 57Co −3/2 271.79 d e142 49Ti −5/2 Stable 166 56Ni 0 6.077 d e+,�+145 50Ti −3 Stable 169 57Ni −1/2 35.60 h e+,�+

170 58Ni −1 Stable e+,�+

occurs in the absorption and fragmentation cross sections. Herebecause of the re-assignment of the mass number when usinga reduced grid, an error is introduced by the change in neutronnumber from its true values. The fluence spectra of high-energyneutrons produced in fragmentation events is also modified byforcing the physics onto a reduced isotopic-grid. This lattererror is expected to be small for heavy target nuclei (A > 16)

because neutron production is dominated by light-particle(n, p, d, t, h, and �) interactions on target nuclei, but may benon-negligible for light target atoms (A�16) (Cucinotta et al.,1998). Since materials with high-hydrogen content are knownto be the optimal shielding materials, the changes in the neu-tron fluence due to the use of a full isotopic-grid should beconsidered.

4. Quantum fragmentation model

We next discuss the quantum multiple scattering descriptionof heavy ion fragmentation (QMSFRG), which has been quitesuccessful in describing the physics of the abrasion—ablationmodel of fragmentation and experimental data (Cucinotta et al.1992,1997a,b,1998; Cucinotta and Dubey, 1994). For inclusivereactions where a single fragment originating in the projec-tile is measured, closure is performed on the final target statewith a momentum vector denoted pX used to represent these

states. The reaction is assumed to precede through the abrasionstep producing a pre-fragment, F ∗, and fireball piece from theprojectile–target overlap denoted, R, followed by the ablationstep where the final projectile fragment, F, is formed after nu-clear de-excitation. The total momentum transfer is q=pT −pX

where pT is the initial target momentum. The pre-fragment,F ∗, excitation spectrum following nucleon or alpha particleabrasion can be represented in terms of an impact parameterdependent convolution of the pre-fragment excitation responsefor a transition of the pre-fragment core from state n to n′and the project fireball response (Cucinotta and Dubey, 1994;Cucinotta et al., 1998)

d�

dEF ∗=

⟨T

∣∣∣∣∫

d2qd2bd2b′eiq(b−b′)Pn,n′

× (b, b′, EF ∗) n,n′(q, b, b, EF ∗)

∣∣∣∣ T⟩

, (8)

where b(b′) is the impact parameter, and q the momentumtransfer. The abrasion response is defined as the interaction ofthe projectile fireball, R with the target, T, after performingclosure over the final fireball states (denoted |R〉)

n,n′(q, b, b′, EF ∗)=∫

dkR

(2�)3〈R′|Q+RT (b′)|R〉

× 〈R|QRT (b)|R′〉�(Ei − Ef ), (9)


101 102 103 104 105

φ (E

), 1

/(cm

2 M

eV/u

yr)

100

101

102

103

104

105

106

1H2H3He4He

101 102 103 104 105

φ (E

), 1

/(cm

2 M

eV/u

yr)

10-3

10-2

10-1

100

101

102

20Ne21Ne+22Ne28Si29Si+30Si56Fe54Fe+55Fe+57Fe

E, MeV/u

Solar Min.Solar Max.

E, MeV/u

(a)

(b)

Fig. 2. Energy spectra for (a) hydrogen and helium isotopes near solarminimum (� = 428 MV) and solar maximum (� = 1050 MV); (b) Ne, Si,and Fe isotopes near solar minimum (� = 428 MV) showing contributionsfrom different isotopes to primary GCR composition.

where the QRT represent the fireball-target profile operator,and kR the projectile fireball momentum vector. The abrasion-response represents a complicated many-body operator that issolved by approximation using closure over the target and fire-ball states for evaluating the pre-fragment distribution. The one-particle abrasion response has been evaluated using the shellmodel response functions (Cucinotta and Dubey, 1993). Thepre-fragment excitation is described in terms of the transitionmatrix

Pn,n′(b, b′, EF ∗)= 〈F ∗n |Q+F ∗T (b′)|F ∗n′ 〉× 〈F ∗n′ |QF ∗T (b)|F ∗n 〉, (10)

where matrix elements for the pre-fragment excitation are eval-uated over the many-body profile operators, QF ∗T . In the modela convolution approach is used to derive the mutli-knockoutspectrum from the single-fragmentation term (Cucinotta andDubey, 1994). The resulting excitation spectrum is broad witha shape similar to a log-normal distribution with mean ener-gies from 20 to 30 MeV for one-nucleon removal (Cucinottaand Dubey, 1993, 1994).

The de-excitation of the pre-fragments in nuclear ablation isdescribed in a stochastic process using a Master equation fornuclear de-excitation by particle emission (Cucinotta and Wil-son, 1996). If fb(E, t) is the probability of finding the nuclei bat time t with excitation energy Eb and Pb,k(E) be the proba-

bility that the nuclei, b will emit ion k with energy E, then theMaster equation is

dfb(E∗b , t)

dt=

∑j

∫dEf a(E

∗a , t)Pa,j (E)

−∑

k

∫dEf b(E

∗b , t)Pb,k(E). (11)

In Eq. (11) the first-term on the right corresponds to gains bydecays of ion a emitting ion j to form ion, b, and the secondterm from losses due to decays of ion b into ion c by emittingion k where the j (or k) are light-particle emissions (n, p, d, t, h,or �). The probability of finding the nuclei b at time t with E∗bcan be divided into stable and unstable parts depending on thelowest excitation energy of E∗b , denoted min[Sbj ],f (E∗b , t)= g(E∗b , t)+ h(E∗b , t), (12)

where g and h denote the stable and unstable parts of f, respec-tively. As t →∞, we have h→ 0 such that

limt→∞ f (E∗b , t)= g(E∗b ,∞). (13)

The probabilities for a single-step decay of nucleus a, are de-fined as

Fa,j =∫ E∗−Sa,j

0Pa,j (E) dE, (14)

with a stable daughter nuclei formed corresponding to the prob-ability

Ga,j =∫ E∗−Sa,j

E∗−Sa,j−min[Sb,k]Pa,j (E) dE (15)

and an unstable daughter nuclei formed with probability

Ha,j =∫ E∗−Sa,j−min[Sb,k]

0Pa,j (E) dE. (16)

These probabilities obey the conditions

Fa,j =Ga,j +Ha,j (17)

and∑j

Fa,j = 1. (18)

The integral Eq.(11) can then separated into two parts by usingthe above definitions

gb(E∗b ,∞)

= gb(E∗b , 0)+

∑j

∫ ∞0

dt

∫dEf a(E

∗b , t)Pa,j (E) (19)

and

0= hb(E∗b )+

∑j

∫ ∞0

dt

∫dEf a(E

∗b , t)Pa,j (E). (20)

In evaluating fragmentation cross sections, Eqs. (19) and (20)are solved by iteration up to excitation energies of 200 MeV


(Cucinotta and Wilson, 1996; Cucinotta et al., 1997a), and usingthe approximation of Campi and Hunfner (1981) at higher val-ues. The use of the statistical decay model with accurate nuclearlevel densities that include nuclear shell structure effects and theuse of measured values for the nuclear masses is important foraccurately predicting the odd–even effects in the fragment spec-trum. Previously this was found to be one of the major deficien-cies in other models such as NUCFRG2 (Wilson et al., 1995).

The iso-spin dependence in fragmentation cross sections en-ters in several ways. First, in the many-body profile functions,the energy-dependent two-body nucleon interaction parametersare summed over the possible projectile and target nucleon scat-tering combinations. Second, the nuclear wave-functions aredependent on shell structure. The largest contribution to iso-spin effects occur in the nuclear ablation process (de-excitation)where the pre-fragments formed and their level spectra aregreatly influenced by the projectile or pre-fragment iso-spin.The model uses Coulomb trajectories (Cucinotta et al., 1997a)and in-medium nucleon–nucleon interaction (Tripathi et al.,2001), which are important at lower energies (< 200 MeV/u).

5. Results

We first illustrate the accuracy of the QMSFRG model andthe effects of iso-spin on fragmentation cross sections. Figs. 3

4 6

σ F, m

b

0

25

50

75

100

125

Experiment (Webber et al.)QMSFRG

29Si (0.508 GeV/u)+ H --> ZF

ZF

4

σ F, m

b

0

25

50

75

100

125

28Si (0.508 GeV/u)+ H --> ZF

6 8 10 12 14 16

Fig. 3. Comparisons of the QMSFRG model to experiment for elementalfragment distribution for 28Si and 29Si on H at 0.51 GeV/u. Experimentaldata from Webber et al. (1998).

and 4 show comparisons of the model to experimental datafor the elemental distributions of fragments for several nucleiof similar mass number where nearby isotopes are chosen forcomparison. The data of Webber et al. (1998) used fragmenta-tion of a primary beam and steering magnets to form severalless abundant nuclei, and we note the beams are only about 90%pure for 29Si, 46Sc, and 46Ti results shown. In all comparisonsthe one-nucleon removal cross sections by electro-magneticdissociation are included (Norbury et al., 1988). The iso-spin,Tz = 0 nuclei display large odd–even effects, which are re-duced for the Tz = 0 nuclei. The odd–even effects are presentfor all target nuclei; however, smaller for hydrogen targets dueto the reduced abrasion probability for large mass removal onhydrogen. One of errors that results from transporting ionsusing a reduced mass-grid is seen by comparing fragmentationcross sections for nearby projectiles where large differences inmany of the production cross sections occur for neighboringprojectile nuclei. The model accurately reproduces the effectsobserved in the experiments. Fig. 5 shows comparisons ofQMSFRG predictions to the NUCFRG2 model (Wilson et al.,1995) and to experimental data for 24Mg and 27Al projectilefragments on C targets. The NUCFRG2 model fails to repro-duce the odd–even dependence of the cross sections on thefragment charge; however, this effect is predicted by QMSFRG.Figs. 6–8 show correlations between the model and experi-mental data for 20Ne, 32S, 36Ar, and 40Ar and 56Fe elementalproduction cross sections on several targets and beam energies.The results show good agreement between theory and exper-iments with the QMSFRG model within 25% percent of theexperimental values for over 90% of the data in the literature.

For our extension of the HZETRN code, Table 2 shows theisotopic table of 170 nuclei developed as a complete list ofGCR primary and secondary nuclei to be used in GCR trans-port problems. We have also listed in the Table 2 the iso-spin,and half-life for the unstable nuclei along with the decay mode.This table of nuclei includes all nuclei of significant abundancewith iso-spin +3/2 to −3 that appear in GCR transport prob-lems. By contrast early versions of the HZETRN code used amuch smaller ion table. The expanded grid used here will allowfor improved description of the physics, and to discuss manyapplications where unstable nuclei are central to understanding.Also listed in the first column of Table 2 is the index schemethat is used in the HZETRN code, which is used to label theshielding-depth and energy-dependent fluence matrix. The in-dex for the nuclei are ordered by increasing mass number, A,followed by charge number, Z, for a given A.

For the boundary condition of the GCR, the HZETRN codein the past has considered only a representation of the mostabundant isotope for each charge with the exception of theprevious evaluation of the individual contributions for primarydeuterons and hellions (Cucinotta et al., 1995). Here we con-sider a more accurate boundary condition that re-distributesthe elements into isotopic fractions based on satellite data asdescribed in Eq. (1). The GCR nuclei are completely strippedand therefore the decay mode and half-life for unstable nu-clei could be differential from those observed in laboratorieson Earth. Garcis-Munoz et al. (1987) has noted the following


ZF

8 10 12 14 16 18 20

σ F, m

b

0

40

80

120

160

ZF

10 12 14 16 18 20

ZF

12 14 16 18 20 22

ZF

10 12 14 16 18 20

σ F, m

b

0

30

60

90

120

150

18046Ti (0.57 GeV/u)+ H --> ZF

46Sc (0.57 GeV/u)+ H --> ZF

40Ar (352 MeV/u)+ H --> ZF40Ca (763 MeV/u)+ H --> ZF

(b)

(a)

Fig. 4. Comparisons of the QMSFRG model to experiment for fragmentation of: (a) (left) 40Ar on H at 0.35 GeV/u and for (right) 40Ca on H at 0.76 GeV/u(Knott et al., 1996); (b) (left) 46Sc on H at 0.57 GeV/u and for (right) 46Ti on H at 0.57 GeV/u (Webber et al., 1998).

differences: (1) electron capture branches, which are inactivefor GCR nuclei relative to �-decay, (2) for non-stripped nucleithere will be two S-shell electrons that participate in electroncapture that will not be available in the decay of GCR nuclei;and (3) changes in screening effects. They have made estimatesof the elongation of the half-time for fully stripped nuclei dueto these processes, which indicate an approximate doubling ofthe decay time observed for laboratory nuclei that normallydecay by electron capture. Since these decay times are muchlonger than the transit time of nuclei in shielding they are notconsidered here. However, it will useful in the future to furtherconsider these processes when studying the effects of stoppingGCR nuclei on planetary atmospheres or surfaces and in tissues.

Fig. 9 shows results from the HZETRN code at solar min-imum (� = 428 MV) behind 5 g/cm2 of aluminum shielding.Comparison of the mass–flux spectra for a 59-isotope grid andthe 170-isotope grid are shown. The 170-isotope grid was de-veloped by considering the fragmentation cross sections for alarge number of GCR primary nuclei and dominant fragmentsin several materials. In Fig. 9 we have scaled the flux by thesquare of the ion charge as a measure of the ionization powerof each mass group. Large differences are seen for many nu-clei. Fig. 10a shows the percent error in the mass-fluence spec-tra resulting from the use of the reduced grid for depths of 5and 20 g/cm2 of aluminum. Errors greater than 100% are seenfor many nuclei; however, in most cases such large errors only


occur for the less abundant nuclei. The probability of biologicaleffects are expected to increase in a manner proportional to Z2

for a given energy, and the elemental-flux distribution may bea sufficient test of transport models for supporting exploration

σ F, m

b

0

40

80

120

160

200

Experiment (Webber et al.)QMSFRGNUCFRG2

4 8

σ F, m

b

0

40

80

120

160

200

24Mg(1455 MeV/u)+ 12C --> ZF

27Al (582 MeV/u)+ 12C --> ZF

ZF

12 146 10

Fig. 5. Comparisons of the QMSFRG model to NUCFRG2 and experimentfor elemental cross section for 24Mg on 12C at 1.455 GeV/u (top panel) and27Al on 12C at 0.6 GeV/u. Experimental data from Webber et al. (1990b).

σF (measured), mb

0 50 100 150 200 250 300

σ F (

QM

SF

RG

), m

b

0

50

100

150

200

250

300

H (0.6 GeV/u)CAlCuSnOne-to-One+25 %

σF (measured), mb

0 100 150 200 250 300

σ F (

QM

SF

RG

), m

b

0

100

50

150

200

250

300

H (0.7 GeV/u)CAl (1.2 GeV/u)CuAg (0.7 GeV/u)One-to-One+25 %

50

(a) (b)

Fig. 6. (a) Comparison of QMSFRG to data of Zeitlin et al. (2001) for 20Ne fragmentation on several targets at 0.6 GeV/u; (b) comparison to data ofBrechtmann et al. (1997) for 32S fragmentation on several targets at 0.7 or 1.2 GeV/u.

studies. In Fig. 10b we show a similar comparison to that ofFig. 10a; however, here for the elemental-fluence spectra. Theerrors are indeed less substantial than those of the mass-fluencespectra, yet are larger than 10% in several cases. Similar com-parisons near solar maximum conditions are shown in Figs. 11.In Table 3, results for the elemental and neutron excess depen-dence of the point dose equivalent behind aluminum shieldingare shown. The nuclei with neutron excess Y < − 1 are notsignificant, while all other cases make important contributions.Table 4 shows the fluence at several depths for several of thecosmic-ray clock nuclei and other less abundant nuclei of in-terest for scientific studies.

6. Discussion

It has been recognized for many years that for the descrip-tion of GCR transport in shielding, theoretical models and ex-perimental data describing the nuclear interactions and prop-agation of protons, heavy ions and their secondaries, lead-ing to accurate and computational efficient transport codes areneeded. In the last 25 years such descriptions have improveddramatically with major milestones that include the develop-ment of an accurate free space GCR model (Badhwar andO’Neill, 1992), the HZETRN code (Wilson, 1977; Wilson et al.,1990), the measurement of a significant number of fragmen-tation cross sections (Brechtmann and Heinrich, 1988; Web-ber et al., 1990b; Knott et al., 1996, 1997; Zeitlin et al., 1997,2001), and the development of an accurate nuclear fragmenta-tion model (Cucinotta et al., 1998). Laboratory (Schimmerlinget al., 1989), and spaceflight (Badhwar and Cucinotta, 2000)validation data have also become available during this timeperiod. The combination of the GCR model of Badhwar andO’Neill, QMSFRG cross section data base, and HZETRN trans-port code have been shown to agree with flight measurements ofGCR dose and dose equivalent within ±15% on several space


0 75 150 225 300 375

σ F (

QM

SF

RG

), m

b

0

70

140

210

280

350

H CAl CuAgPbOne-to-One+25 %

0 75 150 225 300 375 450

36Ar + AT -> ZF40Ar + AT -> ZF

σF (measured), mb σF (measured), mb

Fig. 7. Comparison of QMSFRG to data of Iancu et al. (2005) for 36Ar or 40Ar fragmentation on several targets at 0.34 GeV/u.

102102102

σ F (

QM

SF

RG

), m

b

102

H CAl CuPbOne-to-One+25 %

σF (measured), mb σF (measured), mb σF (measured), mb

0.65 GeV/u 1.05 GeV/u 1.6 GeV/u

Fig. 8. Comparison of QMSFRG model to experimental data for 56Fe fragmentation on several targets at 0.65 GeV/u (Flesch et al., 1999), 1.05 GeV/u (Zeitlinet al., 1997), and 1.6 GeV/u (Cummings et al., 1990).

Mass Number, A

5 20 25 30 35 40 45 50 55 60

Z2

x Φ

(A),

1/ (

cm2

yr)

103

104

105

106

107

108

109

59 isotope grid170 isotope grid

10 15

Solar Min.vehicles. However, further spectral data sets, both in space andat heavy ion accelerators, are needed to fully validate thesecodes.

The implementation of heavy ion transport models has pro-gressed from models that did not satisfy unitarity (Letaw et al.,1983), to the current fully energy-dependent models with ac-curate absorption cross sections (Shinn et al., 1994; Wilsonet al., 1993; Cucinotta, 1993). Future work may still be requiredfor light-particle transport (n, p, d, t, h, �, and mesons andtheir decays), including establishing production cross section←−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−Fig. 9. Comparisons of results from the HZETRN Code for the mass fluxdistribution behind 5 g/cm2 of aluminum shielding for solar minimum condi-tions comparing transport with the reduced 59-isotope grid to transport witha full 170-isotope grid (proton fluence shown for A= 1).


0 10 20 30 40 50 60

% E

rror

in M

ass

Flu

ence

from

Isot

opic

Grid

-150

-125

-100

-75

-50

-25

0

25

50

75

100

125

1505 g/cm2 of aluminum20 g/cm2 of aluminum

0 5 10 15 20 25 30

%E

rror

in E

lem

enta

l Flu

x fr

om Is

otop

ic G

rid

0

10

20

5 g/cm2 Aluminum20 g/cm2 Aluminum

Charge Number, Z

-10

-20

Mass Number, A

Solar Min.

(a)

(b)

Solar Min.

Fig. 10. Comparisons of the error that results from the HZETRN Code: (a)the mass fluence distribution and (b) elemental fluence distribution near solarminimum when using a reduced 59-isotope grid compared to transport witha 170-isotope grid.

models and data, and understanding the role of angular deflec-tions, which are more important for neutrons than for heavy iontransport. However, the heavy ion problem is in much bettershape with many of the remaining task ones of implementation.One exception may be improvements in fragmentation crosssections and laboratory validation for Z=1–5 nuclei producedfrom the heavier projectile nuclei (Z > 10).

The present paper addressed two implementation tasks; theuse of a free-space GCR model, which includes the isotopiccomposition of the primaries, and the extension of the HZETRNcode to a complete isotopic-grid. The problem of the isotopicdistribution of the primary GCR and their modulation during thesolar cycle has been treated in a parametric way in this paper.In this approach, we maintain the accuracy of the GCR modu-lation model for the GCR elemental spectra from Badhwar andO’Neill (1992); however, re-distribute the fluence of each ele-ment amongst its isotopes using estimates from satellite data.The coupling of the HZETRN code to the Leaky Box Modelis a possible approach to a more exact treatment of this prob-lem including the description of the energy dependence of the

Mass Number, A

0 10 20 30 40 50 60

% E

rror

in M

ass

Flu

ence

from

Isot

opic

Grid

-150-125-100-75-50-25

0255075

100125150

5 g/cm2 of aluminum20 g/cm2 of aluminum

0 5 10 15 20 25 30

% E

rror

in E

lem

enta

l Flu

x fr

omIs

otop

ic G

rid

-20

-10

0

10

20

5 g/cm2 Aluminum20 g/cm2 Aluminum

Charge Number, Z

Solar Max.

Solar Max.

(a)

(b)

Fig. 11. Comparisons of the error that results from the HZETRN code for(a) the mass fluence distribution and (b) elemental fluence distribution nearsolar maximum when using a reduced 59-isotope grid compared to transportwith a170-isotope grid.

isotopic fractions. However, for shielding applications these ef-fects are expected to be small based on the current study. Theuse of the complete isotopic grid of nuclei allows HZETRNapplications on the study of the so-called cosmic ray “clocks’’with lifetimes similar to the time spent by GCR nuclei in thegalaxy (∼ 1 M yr) as well as studies of signature nuclei fromthe decay of GCR nuclei in the Mars atmosphere or planetarysurfaces. Because the use of a reduced-grid leads to error andthere are no practical limitations in using a complete isotopic-grid at this time, we recommend it be used when initiatives todesign space exploration vehicles begin. Future tasks that re-main are to implement physical models of the GCR isotopicenvironment and to continue to refine the QMSFRG model in-cluding comparisons to new fragmentation data as they becomeavailable. For the many deformed and highly deformed pro-jectile nuclei considered herein, methods to consider this de-formation in the abrasion process are needed. Other consider-ations for future work is to study the isotopic effects for othermaterials and to consider the decay of radioactive isotopes pro-duced as GCR secondaries in the Mars atmosphere, shieldingmaterials or tissue.


Table 3Elemental (Z) and neutron excess (Y ) dependence on GCR dose equivalent behind (a) 0 g/cm2, (b) 5 g/cm2 and (c) 20 g/cm2 of aluminum shielding

Z Y < 0 Y = 0 Y = 1 Y = 2 Y = 3 Y > 3 Total Z

GCR dose equivalent per year for near solar minimum(a)

0 0.00 0.00 0.00 0.00 0.00 0.00 0.001 9.15 0.21 0.00 0.00 0.00 0.00 9.372 0.50 3.61 0.00 0.00 0.00 0.00 4.113 0.00 0.04 0.04 0.00 0.00 0.00 0.084 0.05 0.00 0.03 0.01 0.00 0.00 0.105 0.00 0.23 0.36 0.00 0.00 0.00 0.606 0.00 3.72 0.31 0.00 0.00 0.00 4.037 0.00 0.74 0.99 0.00 0.00 0.00 1.738 0.00 11.26 0.32 0.32 0.00 0.00 11.909 0.00 0.00 0.38 0.00 0.00 0.00 0.38

10 0.00 2.33 0.45 1.48 0.00 0.00 4.2611 0.00 0.00 1.33 0.00 0.00 0.00 1.3312 0.00 6.89 1.86 1.88 0.00 0.00 10.6313 0.00 0.12 2.34 0.00 0.00 0.00 2.4614 0.00 11.22 1.08 1.06 0.00 0.00 13.3615 0.00 0.00 0.85 0.00 0.00 0.00 0.8516 0.00 2.87 0.59 0.65 0.00 0.00 4.1117 0.00 0.00 0.52 0.41 0.26 0.00 1.1918 0.00 1.37 0.06 0.68 0.00 0.05 2.1519 0.00 0.00 1.71 0.00 0.00 0.00 1.7120 0.00 1.57 0.00 0.78 0.78 0.78 3.9221 0.00 0.00 0.00 0.00 1.00 0.00 1.0022 0.00 0.00 0.00 1.02 1.17 1.59 3.7823 0.00 0.00 0.00 0.00 1.00 0.89 1.8824 0.00 0.00 0.00 1.02 1.02 1.89 3.9325 0.00 0.00 0.00 0.00 1.26 1.67 2.9226 0.00 0.00 0.00 2.05 2.26 22.50 26.8027 0.00 0.00 0.00 0.00 0.10 0.00 0.1028 0.00 0.00 0.00 1.45 0.00 0.00 1.45

Total Y 9.70 46.18 13.22 12.81 8.85 29.35 120.11

(b)0 0.00 0.00 0.65 0.00 0.00 0.00 0.651 15.03 2.56 1.74 0.00 0.00 0.00 19.332 2.48 15.60 0.00 0.00 0.00 0.00 18.083 0.00 0.04 0.04 0.00 0.00 0.00 0.094 0.05 0.00 0.04 0.02 0.00 0.00 0.125 0.00 0.23 0.34 0.00 0.00 0.00 0.586 0.03 3.14 0.29 0.01 0.00 0.00 3.477 0.01 0.62 0.83 0.01 0.00 0.00 1.488 0.07 8.67 0.27 0.27 0.00 0.00 9.289 0.01 0.03 0.31 0.01 0.01 0.00 0.37

10 0.03 1.69 0.37 1.10 0.01 0.00 3.2011 0.01 0.06 0.99 0.04 0.01 0.00 1.1112 0.05 4.71 1.33 1.34 0.01 0.00 7.4413 0.01 0.15 1.66 0.03 0.01 0.00 1.8714 0.06 7.46 0.77 0.75 0.01 0.00 9.0615 0.00 0.02 0.60 0.04 0.02 0.01 0.7016 0.02 1.86 0.42 0.48 0.02 0.01 2.8217 0.00 0.02 0.38 0.32 0.21 0.01 0.9318 0.01 0.89 0.08 0.51 0.04 0.06 1.5919 0.00 0.02 1.12 0.04 0.04 0.04 1.2720 0.01 1.03 0.03 0.58 0.57 0.58 2.8121 0.00 0.00 0.02 0.05 0.70 0.08 0.8522 0.00 0.01 0.03 0.72 0.81 1.12 2.6923 0.00 0.00 0.02 0.05 0.71 0.66 1.4424 0.00 0.00 0.03 0.68 0.73 1.33 2.7725 0.00 0.00 0.02 0.05 0.88 1.23 2.1826 0.00 0.00 0.03 1.34 1.50 13.90 16.7727 0.00 0.00 0.00 0.01 0.06 0.00 0.0828 0.00 0.00 0.01 0.87 0.00 0.00 0.88

Total Y 17.88 48.84 12.45 9.30 6.37 19.03 113.88


Table 3 (continued)

Z Y < 0 Y = 0 Y = 1 Y = 2 Y = 3 Y > 3 Total Z

(c)0 0.00 0.00 2.00 0.00 0.00 0.00 2.001 18.95 3.09 2.21 0.00 0.00 0.00 24.242 2.67 17.03 0.00 0.00 0.00 0.00 19.693 0.00 0.03 0.03 0.00 0.00 0.00 0.074 0.04 0.00 0.04 0.02 0.00 0.00 0.105 0.00 0.13 0.21 0.00 0.00 0.00 0.356 0.05 1.51 0.13 0.02 0.00 0.00 1.727 0.02 0.41 0.38 0.01 0.00 0.00 0.838 0.11 3.69 0.12 0.11 0.00 0.00 4.029 0.01 0.04 0.17 0.02 0.01 0.00 0.26

10 0.04 0.75 0.16 0.46 0.01 0.00 1.4311 0.01 0.09 0.45 0.05 0.02 0.00 0.6212 0.08 1.84 0.52 0.50 0.01 0.00 2.9513 0.02 0.12 0.69 0.04 0.02 0.00 0.8914 0.08 2.57 0.29 0.26 0.02 0.01 3.2215 0.01 0.04 0.26 0.05 0.03 0.01 0.3916 0.03 0.71 0.15 0.18 0.03 0.02 1.1217 0.00 0.02 0.19 0.12 0.08 0.02 0.4418 0.01 0.34 0.07 0.19 0.05 0.04 0.7019 0.00 0.03 0.39 0.06 0.05 0.05 0.5820 0.02 0.47 0.05 0.20 0.19 0.19 1.1121 0.00 0.01 0.03 0.07 0.29 0.10 0.4922 0.00 0.01 0.04 0.30 0.33 0.46 1.1423 0.00 0.00 0.03 0.06 0.30 0.31 0.7124 0.00 0.01 0.04 0.26 0.31 0.55 1.1625 0.00 0.00 0.02 0.06 0.36 0.55 1.0026 0.00 0.00 0.03 0.43 0.53 4.13 5.1327 0.00 0.00 0.00 0.01 0.02 0.00 0.0428 0.00 0.00 0.01 0.24 0.00 0.00 0.25

Total Y 22.15 32.94 8.89 3.73 2.68 6.45 76.64

Table 4Annual fluence of several unstable GCR nuclei versus depth of aluminumshielding near solar minimum (�= 428 MV)

Nuclei 0 (g/cm2) 2 (g/cm2) 5 (g/cm2) 20 (g/cm2)

Fluence per cm2 per year6He 0.0 9.1× 101 2.1× 102 5.6× 102

8Li 0.0 1.0× 103 2.2× 103 5.4× 103

9Li 0.0 3.0× 101 7.2× 101 1.8× 102

10Be 8.8× 103 9.1× 103 9.3× 103 8.8× 103

14C 0.0 7.1× 102 1.7× 103 3.4× 103

18O 9.9× 103 9.3× 103 8.4× 103 5.1× 103

26Al 5.7× 102 7.6× 102 9.4× 102 1.0× 103

36Cl 8.1× 102 7.7× 102 7.1× 103 4.4× 102

References

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Isotopic dependence of GCR fluence behind shielding

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