on scaling cosmogenic nuclide production rates for altitude and

On scaling cosmogenic nuclide production rates for altitudeand latitude using cosmic-ray measurements

Darin Desilets *, Marek ZredaDepartment of Hydrology and Water Resources, University of Arizona, Tucson, AZ 85721, USA

Received 18 January 2001; received in revised form 2 August 2001; accepted 2 August 2001

Abstract

The wide use of cosmogenic nuclides for dating terrestrial landforms has prompted a renewed interest incharacterizing the spatial distribution of terrestrial cosmic rays. Cosmic-ray measurements from neutron monitors,nuclear emulsions and cloud chambers have played an important role in developing new models for scaling cosmic-rayneutron intensities and, indirectly, cosmogenic production rates. Unfortunately, current scaling models overlook ormisinterpret many of these data. In this paper, we describe factors that must be considered when using neutronmeasurements to determine scaling formulations for production rates of cosmogenic nuclides. Over the past 50 years,the overwhelming majority of nucleon flux measurements have been taken with neutron monitors. However, in order touse these data for scaling spallation reactions, the following factors must be considered: (1) sensitivity of instruments tomuons and to background, (2) instrumental biases in energy sensitivity, (3) solar activity, and (4) the way of orderingcosmic-ray data in the geomagnetic field. Failure to account for these factors can result in discrepancies of as much as7% in neutron attenuation lengths measured at the same location. This magnitude of deviation can result in an error onthe order of 20% in cosmogenic production rates scaled from 4300 m to sea level. The shapes of latitude curves ofnucleon flux also depend on these factors to a measurable extent, thereby causing additional uncertainties incosmogenic production rates. The corrections proposed herein significantly improve our ability to transfer scalingformulations based on neutron measurements to scaling formulations applicable to spallation reactions, and, therefore,constitute an important advance in cosmogenic dating methodology. ß 2001 Elsevier Science B.V. All rights reserved.

Keywords: scale factor; production; rates; cosmogenic elements; neutrons; cosmic rays

1. Introduction

Cosmogenic nuclides produced in situ in terres-trial rocks have been applied to exposure datingsince the mid-1980s, but until recently [1], there

had been no critical review of the commonly usedaltitude and latitude scaling of cosmogenic pro-duction rates derived by Lal [2] and given in[3,4]. Dunai [1] has proposed a major revision to[2^4] based on data from neutron monitors, nu-clear emulsions and cloud chambers. However, asdiscussed by Desilets et al. [5], Dunai's scalingmodel has many similar shortcomings to Lal'smodel. These two models have the followingweaknesses: (1) Cosmic-ray data are ordered ac-cording to parameters that inadequately describe

0012-821X / 01 / $ ^ see front matter ß 2001 Elsevier Science B.V. All rights reserved.PII: S 0 0 1 2 - 8 2 1 X ( 0 1 ) 0 0 4 7 7 - 0

* Corresponding author. Tel. : +1-520-621-4072;Fax: +1-520-621-1422.

E-mail addresses: [email protected] (D. Desilets),[email protected] (M. Zreda).

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Earth and Planetary Science Letters 193 (2001) 213^225

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the geomagnetic shielding e¡ect. Latitude surveydata are ordered according to geomagnetic incli-nation [1] and according to geomagnetic latitude[2]. However, the cosmic-ray intensity varies by asmuch as 15% along both constant geomagneticlatitude and inclination. (2) The e¡ects of solaractivity on latitude and altitude survey data areeither completely neglected [1] or addressed un-clearly [2]. (3) The energy dependence of the nu-cleon attenuation is either underestimated, be-cause of limited data [2], or ignored [1]. (4) Bothmodels are based on a small selection of cosmic-ray data, con¢ned mostly to the 1950s.

In order for the cosmogenic nuclide datingmethod to be successfully applied at di¡erent lo-cations, the altitude and latitude scaling of nucle-on intensity must be accurately constrained. Amore re¢ned scaling model can be derived fromthe numerous latitude and altitude surveys per-formed since the 1950s; however, these datamust be used with caution. The purposes of thispaper are to review some of the major cosmic-raysurveys conducted over the past 50 years and todiscuss how these data can be used to derive anaccurate scaling model. We also review some ba-sic concepts and de¢nitions in cosmic-ray physicsin order to build the framework for these discus-sions.

2. De¢nitions

Primary cosmic rays are charged particles im-pinging on Earth with relativistic energies. Cosmicrays arriving from outside the solar system areknown as galactic cosmic rays (GCRs), whereasthose arriving from the sun are known as solarcosmic rays (SCRs) [6]. Secular variations in solaractivity cause the low-energy (E6 1 GeV) GCR£ux to vary by as much as an order of magnitude,whereas the high-energy GCR £ux (Es 10 GeV)is mostly insensitive to solar activity [7].

Secondary cosmic rays are produced throughthe interaction of primary cosmic rays with at-mospheric and terrestrial nuclei. The secondary£ux includes strongly interacting particles (e.g.neutrons, protons and pions), weakly interactingparticles (e.g. muons), and electromagnetic radia-

tion (e.g. Q and L radiation). Neutrons are respon-sible for the majority of nuclear transformationsnear the Earth's surface [4].

Neutrons may be classi¢ed by energy accordingto the types of nuclear reactions in which they areinvolved. Although there is no standard conven-tion for classifying neutrons, the following de¢ni-tions are useful [8,9].

High-energy neutrons are produced through di-rect reactions of primary and secondary cosmic-ray particles with terrestrial nuclei. In a directreaction, the incident particle interacts separatelywith a small number of individual nucleons, usu-ally at the surface of the nucleus. A high-energyneutron may in turn liberate additional high-en-ergy particles in a chain reaction process. Thesereactions may occur within the nucleus (intranu-clear cascade) or between nuclei (internuclear cas-cade). The de Broglie wavelength of a particle isinversely related to particle momentum, andtherefore at lower momenta, interactions withthe entire nucleus become more probable. Theenergy of the incident particle is then distributedthroughout the entire nucleus in what is known asa compound^nucleus reaction. Spallogenic pro-duction of nuclides may occur from both direct-and compound^nucleus reactions. We de¢ne high-energy neutrons to be those capable of producingspallation reactions, which corresponds to ener-gies ranging from primary energies down to about10 MeV.

Fast neutrons are produced primarily from thede-excitation of nuclei following compound^nu-cleus reactions. A common mode of de-excitationis through the emission of neutrons and protonsaccording to a Maxwellian energy spectrumpeaked at about 1 MeV [10]. This process isknown as nuclear evaporation, by analogy tomolecules evaporating from the surface of aheated liquid. Unlike neutrons produced in directreactions, the energy spectrum and angular distri-bution of evaporation neutrons does not dependcritically on the energy and direction of the ini-tiating particle. In other words, the excited nu-cleus does not `remember' these properties of theinitiating particle. However, as the energy of theincident nucleon increases, the nucleus is morelikely to be excited to a higher `temperature',

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D. Desilets, M. Zreda / Earth and Planetary Science Letters 193 (2001) 213^225214

and the average number of neutrons evaporatedby the nucleus therefore increases. Evaporationreactions may also be induced by particles suchas pions, muons and photons. Fast neutrons gen-erally have insu¤cient energy to produce furtherevaporation reactions, and therefore do not ini-tiate nuclear spallations. The energy range forfast neutrons is de¢ned here to be approximatelyfrom 10 MeV to 100 keV.

Slow neutrons are produced from the slowingdown (`moderation') of fast neutrons, throughelastic and inelastic collisions with nuclei. We de-¢ne slow neutrons to be those with energies onthe order of 1 keV. Thermal and epithermal neu-trons are produced from the slowing down of fastneutrons to energies on the order of the vibra-tional motion of nearby molecules. An importantcharacteristic of thermal and epithermal neutronsis their relatively high probability of being ab-sorbed by nuclei. Thermal neutrons are de¢nedto be in vibrational equilibrium with the mole-cules of the surrounding medium, which at a tem-perature of 293.16 K corresponds to an averageenergy of 0.025 eV. Epithermal neutrons are de-¢ned here as those with energies between 100 eVand the cadmium cuto¡ energy for transparencyto neutrons of 0.5 eV.

3. The nucleon attenuation length

Primary cosmic rays collide with oxygen andnitrogen nuclei near the top of the atmosphere,initiating cascades of protons, neutrons and othersecondary particles (Fig. 1). In the lower atmo-sphere (s 200 g cm32), the nucleon £ux diminishesas a function of mass-shielding depth approxi-mately according to:

N2 � N1 expZ13Z2

1 N

� ��1�

where N1 and N2 are the nucleon £uxes at depthsZ1 and Z2 (g cm32), respectively, and 1N is thenucleon attenuation length (g cm32) (also referredto as the absorption mean free path, absorptionlength [12] or e-folding length). A nucleonic cas-cade loses energy through nuclear collisions and

electromagnetic interactions, and this energy lossdepends on the total mass of air transited by thecascade. The Earth's atmosphere is approximately1033 g cm32 thick and has a density that varieswith altitude, latitude and time. A hypotheticalrelationship between mass-shielding depth and al-titude corresponding to the year-round, mid-lati-tude pressure distribution is given by the US stan-dard atmosphere [13] (Fig. 2). Deviations fromthe US standard atmosphere in areas of statisti-cally high and low pressure, such as the SiberianHigh and the Aleutian Low, should be taken intoaccount when scaling cosmogenic productionrates [14].

Eq. 1 assumes a monodirectional, monoener-getic beam of stable particles that does not initiate

Fig. 1. Propagation of the secondary cascade through the at-mosphere (adapted from [11]).

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chain reactions. The terrestrial cosmic-ray £ux,however, follows an approximately power-law en-ergy spectrum, propagates as a cascade, includesshort-lived particles (e.g. pions and muons), and isdistributed about the zenith roughly according toa cosine function [15]. Nonetheless, Eq. 1 providesa satisfactory description of cosmic-ray nucleonabsorption over small atmospheric depths(V100 g cm32).

Previous workers [1,4] used atmospheric mea-surements of nucleon intensity to derive the alti-tude dependence of spallation reactions. Accord-ing to Dunai [1], Lal's [4] scaling modeloverestimates the value of the atmospheric attenu-ation length for cosmogenic nuclide productionby nucleons (here termed 1prod;N ) at both highand low latitudes. This claim is based on measure-ments from cloud chambers, nuclear emulsionsand neutron monitors [1] that seem to indicatelow-altitude attenuation lengths of 130 g cm32

at high latitudes and 149 g cm32 at low latitudes.Lal's scaling formula gives values of 135 g cm32

and 157 g cm32, respectively, for these locations.

Here, we discuss how instrumental biases a¡ectmeasurements of the nucleon attenuation lengthand how these biases can at least partially recon-cile discrepancies between measured attenuationlengths. We also introduce neutron monitor datafrom extensive altitude and latitude surveys ofnucleon intensity.

3.1. Neutron monitor data

The most comprehensive and well-reported in-vestigation of the global distribution of nucleonintensity is probably the neutron monitor surveyconducted by Carmichael et al. [16^18] during theInternational Quiet Sun Year (1965^1966). Thesemeasurements are ideally suited for scaling neu-tron monitor counting rates because: (1) they cov-er a wide range of geomagnetic cuto¡s (0.5^13.3GV) and atmospheric depths (200^1033 g cm32) ;(2) they have been corrected for secular variationsin the primary cosmic-ray intensity and temper-ature e¡ects ; and (3) they were taken with a land-based neutron monitor and an airborne monitorthat had been cross-calibrated.

Several other surveys of neutron monitor at-tenuation lengths have also been published, butunfortunately the experimental procedures, cor-rections and raw data have not always been re-ported in adequate detail. The only survey ofcomparable scope to [16^18] covered a similarrange of cuto¡s, but was mostly limited to depthsgreater than 880 g cm32 [19]. An earlier survey byBachelet et al. [20] de¢ned the general character-istics of the neutron monitor attenuation length(1NM) as a function of atmospheric depth andcuto¡ rigidity, but was more limited in scope.Neutron monitor attenuation lengths have alsobeen evaluated using data from the world-widenetwork of ¢xed neutron monitors [21]. Othersurveys generally give relationships consistentwith that found by [16^18].

The primary goal of [16^20,22] was to deter-mine the altitude and latitude dependence of1NM so that counting rates of neutron monitorscould be corrected for small temporal variationsin barometric pressure. The survey by [16^18]shows that 1NM reaches a minimum near 850 gcm32 (Fig. 3), in agreement with [19] and [20].

Fig. 2. The relationship between atmospheric depth and alti-tude in the US standard atmosphere [13]. For comparison,relationships found in Antarctica [14] and in Hawaii (ourown data) are shown. The curve for Hawaii is based onglobal positioning system (GPS) and barometer measure-ments (given by open squares) taken on 7^12 April 2000.Barometric measurements are 10 min averages and are there-fore not necessarily representative of long-term or even re-cent pressure conditions on Hawaii.

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Between the top of the atmosphere and 850 gcm32, 1NM decreases with increasing depth asthe overall energy of the cascade dissipates [23].However, beyond this depth 1NM increases withincreasing depth, which is attributable to the com-bined e¡ects of three phenomena [18] : (1) Muonsinteract with the lead portion of a neutron mon-itor to produce neutrons that contribute to thetotal counting rate. Because the muon £ux ismore highly penetrating than the nucleon £ux,1NM at greater atmospheric depths re£ects theincreased proportion of muons contributing tothe counting rate. Muon sensitivity near sea levelappears to be a feature of all neutron monitorsbased on the IGY/NM-64 design [24]. (2) K-Con-tamination of the counter tubes produces a con-

stant background amounting to about 1% of thehigh-latitude sea level counting rate. (3) Cosmic-ray nucleons incident from oblique angles are se-lectively ¢ltered with depth in the atmosphere.The omnidirectional nucleon £ux therefore at-tenuates more rapidly with increasing atmosphericdepth than does the vertical nucleon £ux.

For neutron monitor counting rates to be usedfor scaling production rates, muon and back-ground contributions must be removed. Fortu-nately, the relative contributions of muons andneutrons to the neutron monitor counting rateare fairly well known (Table 1). Attenuationlengths for fast muons and slow negative muonsas a function of PC are also fairly well known[15]. Assuming, after [18,19] and based on [25],that slow and fast muon £uxes have similar lati-tude distributions, we calculated 1N from the neu-tron monitor data of [16,17] and the values inTable 1 using the relationship:

11 NM

�

Xn

i�1

1 i

NiXn

i�1

Ni

�1 N

NN� 1 W3�s�

NW3�s�� 1 W �f�

NW �f�� 1 B

NB

NN �NW3�s� �NW �f� �NB�2�

where Ni and 1i are the counting rate and attenu-ation length, respectively, for the ith component.At sea level and high latitude, contributions fromnucleons, slow negative muons, fast muons andconstant background (NN , NW3�s�, NW �f�, and NB,respectively) account for more than 98% of theneutron monitor counting rate.

Attenuation lengths given by Dunai [1] (V130g cm32 at 2 GV) appear to agree reasonably wellwith 1N measured with a neutron monitor (Fig.3). However, 1N is equivalent to 1prod;N only ifeither the nucleon attenuation length is indepen-

Fig. 3. E¡ective attenuation lengths for an NM-64 neutronmonitor based on measurements taken in April^June, 1965[18]. The e¡ective nucleon attenuation length was calculatedby removing the contribution of muons and background to1NM using Table 1 and Eq. 2. The e¡ective attenuationlength reduces the neutron monitor counting rate at somedepth to the sea level counting rate.

Table 1Contributions to the NM-64 neutron monitor counting rate [24] and attenuation lengths for fast and slow muons [15]

2 GV 13 GV

Contribution 1 Contribution 1(%) (g cm32) (%) (g cm32)

Slow muons 3.6 þ 0.7 240 6.0 281Fast muons 2.0 þ 0.4 560 3.3 640Background 1.0 r 1.8 r

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dent of energy (the shape of the nucleon energyspectrum is invariant with depth) or the energysensitivity of the measuring apparatus is identicalin form to the energy dependence of nuclide pro-duction. In Sections 3.2^3.4 we show that, strictlyspeaking, neither assumption is valid.

3.2. Energy dependence of the nucleon attenuationlength

Data from neutron multiplicity counters [26^28]demonstrate that 1N decreases with median nucle-on energy. A multiplicity counter is a neutronmonitor that records the number of countingevents occurring within a gating time of about700^1000 ms [27,29]. Each high-energy nucleoninteracting with the monitor produces about1.44 counts, on average, at high latitude and sealevel [29]. However, the number of counts result-ing from each interaction depends on the energyof the incoming nucleon. This is because a neu-tron monitor records evaporation neutrons, andthe average number of evaporation neutrons pro-duced in a reaction increases with the increasingenergy of the interacting nucleons. A neutronmultiplicity counter, therefore, provides informa-tion on the energy spectrum of secondary cosmic-ray nucleons [29]. Fig. 4 demonstrates that theattenuation length decreases with increasing me-dian nucleon energy, at least in the range 120^700MeV. This behavior is a consequence of the factthat the lower energy nucleons in a particle cas-cade are produced (both directly and indirectly)from nucleons of higher energy, and since no cos-mic-ray e¡ect can decrease with increasing depthany faster than the primary radiation from whichit originates [30] the attenuation length must de-crease or remain constant with increasing nucleonenergy.

In contrast to Dunai [1], Lal [2] explicitly rec-ognized the energy dependence of 1N in derivinghis scaling model. However, prior to [27], 1N wascommonly assumed to be constant with energy upto about 400 MeV. This assumption was based onearly experiments and simpli¢ed cascade models[2,10,12,27], despite other experimental evidenceto the contrary [30^32]. Under the assumptionthat the slow neutron £ux is proportional to the

high-energy nucleon £ux for energies below 400MeV, Lal [2] based a major portion of his scalingon airborne measurements of slow neutron £uxes[33]. Recognizing that for some reactions a signif-icant portion of cosmogenic nuclide productionoccurs at energies above 400 MeV, and that atthese higher energies the nucleon attenuationlength decreases with energy, Lal [2] applied cor-rections that tended to decrease attenuationlengths derived from slow neutron measurements.Hence, Lal [2^4] gives scaling factors for bothslow neutron £uxes and for the total nuclear dis-integration rate in the atmosphere. Although thecorrection applied by Lal [2] is approximate andfails to account for the energy dependence of 1N

below 400 MeV, Lal considered that the altitudedependence of a particular reaction may dependon the energy at which that reaction occurs. Fu-ture work should also consider this energy depen-dence.

3.3. Energy sensitivity of neutron monitors

Because high-energy nucleons produce moreneutron monitor counts per interaction than

Fig. 4. Neutron monitor attenuation length (corrected formuon and background contributions) as a function of neu-tron multiplicity. Measurements taken between 952 and 544g cm32 at 2 GV with an IGY type monitor [28]. Median nu-cleon energies are based on calculations for a high-latitudesea level IGY neutron monitor [24].

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low-energy nucleons, the neutron monitor re-sponse is biased towards the higher end of thenucleon energy spectrum. An additional e¡ect isproduced by 7.5 to 28 cm of para¤n or polyeth-ylene (known as the `re£ector') on the outside ofthe monitor which substantially modulates the£ux of nucleons with E6 50 MeV. Hughes andMarsden [29], for example, showed that an IGYneutron monitor records a negligible number ofcounts from nucleons with E6 50 MeV. Theseconsiderations imply that for nuclide productionat low thresholds (V20 MeV), such as K(n,x)36Cland Ca(n,x)36Cl reactions [15], 1N measured froma neutron monitor may underestimate 1prod;N ,and, therefore, a correction may be needed inthe direction opposite to that applied by [2] toslow neutron data. This correction should bemost important for neutron monitors with thickre£ectors, such as the IGY type, and less impor-tant for those with thin re£ectors, such as theNM-64 type. On the other hand, for the reactionO(n,x)10Be produced by neutrons having a me-dian energy (Emed) of V140 MeV at high latitudeand sea level, the neutron monitor response(Emed = 130^160 MeV) may accurately describethe altitude dependence of cosmogenic nuclideproduction [15].

3.4. Energy sensitivity of cloud chambers andemulsions

Cloud chambers and nuclear emulsions recordthe tracks of ionizing particles produced in nu-clear disintegrations (or `stars') initiated by cos-mic rays. Each track is known as a prong, and thenumber of prongs (or `star size') produced in adisintegration is proportional to the kinetic energyof the disintegration-producing particle. Althoughthe precise relationship between incident nucleonenergy and prong number is somewhat ambigu-ous, it is clear from theoretical considerations [34]that mean star size increases monotonically withincreasing nucleon energy. Above sea level, moststars recorded in emulsions are initiated by pro-tons and neutrons, while the contribution fromslow muons is negligible [35].

Due to the undercounting of one- and two-prong stars, nucleon attenuation lengths measured

with cloud chambers and nuclear emulsions aretypically biased towards high energies. Under-counting occurs because tracks left by one- andtwo-prong stars are di¤cult to distinguish fromtracks left by proton recoils and scattering events.Also, there is generally a low scanning e¤ciencyfor one- and two-prong stars [2]. For example,Brown [32] undercounted one- and two-prongstars occurring in a cloud chamber, while Dixit[36] and Roederer [31] neglected these altogether.In the silver bromide emulsions used by [36] and[31], one- and two-prong stars correspond to en-ergies of about 41 MeV and 90 MeV, respectively[34]. As with neutron monitors, the e¡ect of en-ergy bias is to underestimate the total nucleonattenuation length. Correcting the data of [32]for this e¡ect using the procedure outlined byLal [2] yields a value of 137 þ 5 g cm32 for the£ux-weighted attenuation length of nucleons withEs 40 MeV between 700 and 1032 g cm32. Thiscompares to the value of 132 þ 4 g cm32 originallygiven by [32] for the uncorrected nucleon attenu-ation length between these depths. Failure tomake corrections such as this can account for ap-parent discrepancies between attenuation lengths.

4. The geomagnetic e¡ect and cuto¡ rigidity

The geomagnetic ¢eld imposes a rigidity (mo-mentum-to-charge ratio) cuto¡ on primary cos-mic-rays. The value of this cuto¡ tends to increasewith decreasing latitude, resulting in lower cos-mic-ray intensity towards the equator. However,only in a centered dipole ¢eld is the cosmic-rayintensity a unique function of geomagnetic lati-tude. In a magnetic ¢eld with substantial non-di-pole components, such as the present geomagnetic¢eld, there is also a `longitude e¡ect' in cosmic-ray intensity. Data from so-called `latitude sur-veys' must therefore be ordered in terms of a pa-rameter that accounts for both latitude and lon-gitude e¡ects.

Prior to the mid-1950s most cosmic-ray mea-surements were ordered according to geomagneticlatitude (Vm) calculated from a centered dipolerepresentation of the geomagnetic ¢eld [8]. How-ever, the ¢rst latitude surveys [8,23] proved that

EPSL 5979 14-11-01


non-dipole components have a substantial e¡ecton the cosmic-ray intensity. Early investigatorsalso attempted to characterize primary cosmic-ray access by using the lower vertical cuto¡ ri-gidity (PL) based on a dipole representation ofthe geomagnetic ¢eld [37] :

PL�V m� � 30 M0

4 r2 cos4 V m � 14:9 GVMM0

� �cos4 V m

�3�

where M0 is a reference dipole moment(7.90622U1022 A m2 for Epoch 1980.0 [37]), Mis the dipole moment at some other time and r isthe average radius of the Earth (6.371U106 m[37]). A coe¤cient of 14.9 GV, based on the mag-netic survey of 1944 [7], is often given in Eq. 3.

Eq. 3 gives the minimum momentum-to-chargeratio that a vertically incident primary cosmic-rayparticle must possess in order to gain access to acertain geomagnetic latitude (Vm) in a centereddipole ¢eld. Besides the limitation that Eq. 3only applies to a centered dipole ¢eld, there isthe additional shortcoming that it implicitlyignores the presence of the solid Earth. Conse-quently, particles with rigidities above the lowercuto¡ may still fail to reach latitudes permitted byEq. 3, since these particles may travel along com-plex trajectories which intersect the Earth else-where [39]. Hence, there also exists an upper cut-o¡ rigidity, PU, above which all primary particlesare accepted. The region between the lower and

upper cuto¡ is termed the penumbra, and hereacceptance or rejection of primary cosmic-raysdepends on individual particle trajectories. Inthe 1950s and early 1960s, several investigators[40^42] derived analytical and semi-empiricalequations for cuto¡ rigidity in attempt to at leastpartially account for non-dipole and penumbrale¡ects. Although these e¡orts led to successiveimprovements in ordering cosmic-ray data fromlatitude surveys, satisfactory cuto¡s did not ariseuntil computer-intensive calculations ¢rst becamepractical in the middle 1960s [43].

Since the late 1960s, most neutron monitormeasurements have been ordered in terms of ef-fective vertical cuto¡ rigidity (PC) (Fig. 5) calcu-lated by integrating cosmic-ray trajectoriesthrough a high-order mathematical model of thegeomagnetic ¢eld [44^47]. This method involvestracing the paths of negatively charged particlesas they are leaving speci¢c locations above theEarth in the vertical direction. Anti-protons thatescape the geomagnetic ¢eld to in¢nity followpaths identical to those of cosmic-ray protons in-cident on the same location from in¢nity. In orderto locate the lower, upper and e¡ective cuto¡s, alarge number of trajectories corresponding to arange of particle energies must be numericallytraced.

The primary £ux is nearly omnidirectional andtherefore a complete description of primary cos-mic-ray access to the Earth requires calculation ofcuto¡ rigidities for all angles of incidence [48].

Fig. 5. The world-wide distribution of PC (GV) for Epoch 1955.0 [44].

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However, because vertically incident primariesproduce most of the sea level nucleon £ux, ithas proved adequate in practice to order cosmic-ray data according to cuto¡s calculated only forthe vertical direction. The reliability of PC hasbeen con¢rmed by numerous sea level latitudesurveys which show a smooth and consistent re-lationship between cuto¡ rigidity and nucleon in-tensity [45^47]. Failure to account for obliquelyincident primary particles appears to have onlya minor e¡ect on ordering cosmic-ray data[47,48].

The adoption of PC for cosmic-ray surveys wasa major advance in experimental cosmic-rayphysics. However, the currently available scalingmodels order cosmic-ray data according to geo-magnetic inclination [1] and geomagnetic latitude[2^4], both of which are ine¡ective at describingshielding e¡ects of the prevailing geomagnetic¢eld [5,49]. Most importantly, the geomagnetic¢eld contains a substantial quadrupole compo-nent, which makes the present-day geomagnetic¢eld approximately equivalent to a dipole ¢eldshifted about 392 km from the center of the Earthtowards Southeast Asia [38]. This e¡ect causes sealevel nucleon intensity along the cosmic-ray equa-tor (the line of minimum cosmic-ray intensityaround the Earth) to decrease by about 15%from South America to Southeast Asia. Geomag-netic inclination and geomagnetic latitude cannotpossibly account for this e¡ect.

Direct measurements of the cosmic-ray inten-sity are collected in the present-day geomagnetic¢eld and therefore should be ordered according toPC. Unfortunately, PC cannot be accurately cal-culated for the past 200^10 000 years because thegeomagnetic ¢eld parameters are not known.However, if the long-term (s 10,000^20,000years) behavior of the Earth's magnetic ¢eld canbe approximated by an axial dipole ¢eld, as isoften assumed [1,50] then geomagnetic latitude(Vm) is equivalent to geographic latitude (V) overthe long term. PL can then be calculated from Eq.3, and PC can be estimated from Fig. 6. It shouldbe noted that, strictly speaking, Fig. 6 representsthe penumbral correction corresponding only to acentered dipole model of the recent (past V50years) geomagnetic ¢eld. Deviations caused by

non-dipole characteristics of the geomagnetic ¢eld(Fig. 5) and variations in dipole intensity are notaccounted for.

5. Solar activity

Solar activity substantially reduces the £ux ofGCR particles at high latitudes, but has only asmall e¡ect on the £ux at low latitudes. Duringperiods of high solar activity, the sun emits asubstantial £ux of low-energy SCR protons, en-hancing the overall £ux of low-energy primaryparticles traveling through the interplanetary me-dium. However, associated with these low-energySCR particles are traveling magnetic ¢elds whichscreen the Earth from low-energy GCR particles[7]. The net e¡ect is a decrease in the cosmic-rayintensity reaching the Earth while the sun is ac-tive.

Solar modulations cause the GCR £ux reaching

Fig. 6. The penumbral correction in (A) a centered dipole¢eld [15,41,42,57,58] and (B) a high-order model of the realgeomagnetic ¢eld [43,47].

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the Earth to vary in time according to an 11-yearcycle. Because low-energy primaries are alwaysrejected by the geomagnetic ¢eld at low latitude(high PC), the e¡ects of solar modulation are sig-ni¢cant only at high latitude (low PC). Also, be-cause secondary cascades initiated by low-energyGCRs tend to be weakly penetrating, variationsin secondary intensity due to solar e¡ects becomemore subdued towards sea level.

Several experiments demonstrate that even atdepths ranging from 680 to 1033 g cm32 the shapeof the neutron £ux latitude curve depends consid-erably on solar activity (Fig. 7) [20,50^53]. Fromsolar minimum to solar maximum, the high-lati-tude sea level nucleon £ux decreases by about 8%,whereas at 680 g cm32 the £ux decreases by about21%. At low latitudes (V14 GV), solar modula-tions have a negligible e¡ect on sea level neutronintensity [50], while at 680 g cm32 the neutron£ux varies by only about 5% [28].

The nucleon attenuation length also varies withsolar activity (Fig. 8). This e¡ect is related tochanges in the primary energy spectrum causedby solar modulations. Periods of high solar activ-ity are associated with a lower primary £ux but a

harder primary energy spectrum. Cascades initi-ated by more energetic primaries tend to havecorrespondingly longer attenuation lengths. Dur-ing low solar activity, the overall nucleon attenu-ation length re£ects the increased contribution oflow-energy cascades which attenuate rapidly inthe atmosphere.

6. Conclusions

The development of an accurate model for scal-ing production rates is an essential step towardsre¢ning the cosmogenic nuclide dating method.Improvements to existing scaling models [1,4]could be made by: (1) considering all cosmic-raydata accumulated since the 1950s; (2) using im-proved corrections for instrumental biases (3) or-dering cosmic-ray latitude survey data according

Fig. 8. (A) Attenuation lengths measured from an IGY mon-itor at 2 GV and sea level [60] over one solar cycle. (B) A10-day moving average of the relative counting rate of theClimax IGY neutron monitor over the same period (http://ulysses.uchicago.edu/NeutronMonitor/).

Fig. 7. Latitude surveys of nucleon intensity conducted at so-lar maximum and solar minimum, normalized at 14 GV. Air-borne and sea level curves correspond to atmospheric depthsof 680 g cm32 and 1033 g cm32, respectively [47,52,53,59].

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to PC ; and (4) using more realistic relationshipsbetween atmospheric mass-shielding depth and al-titude.

There appears to be general agreement thatmore work is needed to constrain the altitudeand latitude dependence of 1prod;N [1,5,54^56].We are currently working on this issue. By farthe largest body of data on nucleon intensity isfrom neutron monitors, which require corrections.However, without more accurate knowledge ofnucleon excitation functions and of the energy de-pendence of the nucleon attenuation length, anycorrection to 1NM could potentially carry a largeuncertainty. On the other hand, the direct mea-surement of 1prod;N in geological samples, such aslava £ows that extend from high altitudes to sealevel, requires exceptional ¢eld conditions (easilydistinguishable £ows, evidence of minimal erosionand minimal ash cover) if long-lived nuclides areto be applied. It also seems unlikely that a su¤-cient number of sites can be found to give ad-equate latitude and altitude coverage for develop-ing an accurate scaling model based entirely ongeological samples. Work being done with arti¢-cial targets [55,56] should therefore play an im-portant role in validating scaling models.

Acknowledgements

We thank Nat Lifton and Devendra Lal forvaluable discussions, and Fred Phillips and ananonymous reviewer for their helpful suggestions.This work was supported by National ScienceFoundation Grants EAR-0001191 and ATM-0081403 and Packard Fellowship 95-1832. Wealso acknowledge National Science Foundationsupport of the Climax neutron monitor underGrant ATM-9912341.[RV]

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on scaling cosmogenic nuclide production rates for altitude and

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