Figure 3.1 Fate of a primary charged galactic cosmic ray particle. a. Influence on a proton trajectory of a geomagnetic dipole field that is tilted 22^o from Earth’s spin axis shown as the western half of the field in a meridian plane, compiled from Rossi (1964) and (Quenby and Wenk, 1962). ‘e.p.’ is equatorial plane. According to Stormer theory (see text), outer shaded region is forbidden to particles of a given insufficient rigidity. Trajectories 1, 2, and 3 of rigidity equivalent to the vertical threshold rigidity, arriving from infinity, must pass through the jaws into inner allowed region and ultimately follow a dipole field line down the horn to atmosphere. Trajectory 4 is impossible due to the opacity of the earth. Distribution of allowed main cone ‘a’, Stormer and shadow forbidden cones ‘f’, and the penumbra ‘p’ are approximated from Pomerantz (1971) for a 10 GV positively charged particle at mid-latitude. b. The major components of a cosmic ray extensive air shower (cascade), showing secondary particle production in the atmosphere and rock (modified from Allkofer and Grieder, 1984 and Clay and Dawson, 1997). Particle symbols are given in the notation list. Numbers refer to examples of in situ cosmogenic nuclide interactions: (1) ³⁵Cl(n_th, g)³⁶Cl; (2) ¹⁶O(n, 4p3n)¹⁰Be; (3) ²⁸Si(n,p2n)²⁶Al. Vertical scale not linear.

Figure 3.2 Variation of the production of TCN with depth in rock at high latitude and sea level: (a) ¹⁰Be in quartz arenite; (b) ³⁶Cl in an ultramafic rock. Average ultramafic rock composition from Fabryka-Martin (1988) was used, but Cl concentration was reduced to 10 ppm.

Figure 3.3 (a) Profiles of the thermal and epithermal neutron fluxes with depth in a variety of rock types. Rock compositions were taken from average values given by Fabryka-Martin (1988) and rocks were assumed to be at sea level, high latitude and to be dry. Heavy lines indicate thermal neutron fluxes and thin lines epithermal fluxes. (b) Epithermal neutron flux as a function of depth and water content in an average shale (Fabryka-Martin, 1988). Shale was assumed to have 30% porosity. Numbers beside curves refer to volumetric water content. (c) Thermal neutron flux as a function of depth and water content in same shale.

Figure 3.4 (a) Coordinate system for computations involving cosmic ray distributions. The origin for the angle of inclination, f, is toward the zenith and the origin for the azimuth, q, is north. (b) Illustration of solid geometry used in integration of the cosmic-ray flux.

Figure 3.5 Variation of the cosmic ray intensity (F(f)) with inclination angle (f).The variation of the intensity is shown for both F(f) = F_ocos^2.3f and F(f) = F_ocos^3.5f.

The fraction of the total flux passing through a surface rotated so as to be always normal to the flux, the fraction of the total flux passing through a horizontal surface, and the cumulative fraction of the total flux passing through a horizontal surface are also illustrated, based on F(f) = F_ocos^2.3f.

Figure 3.6. Apparent attenuation length (L_f) versus latitude (l_geo). Solid symbols are rock depth profiles in upper 300 g·cm^-2, open symbols are from measurements in or at the bottom of the atmosphere. All other curves are linear fits on L_f with at least 3 measurements. High latitude sites are assigned to latitude 60°, above which L_f is invariant. Refer to Table 3.

Figure 3.7. The Sint 200 record (dots and thin curve). The relative record was normalized for absolute mean VADM as discussed in text using the absolute paleointensity data (open circles) from (Coe, 1978; Barbetti and Flude, 1979; Kovacheva, 1980; Kovacheva, 1982; McElhinny and Senanayake, 1982; Chauvin et al., 1989; Chauvin et al., 1991; Brassart et al., 1997). Thick solid curve is the Holocene absolute paleointensity from McElhinny and Senanayake (1982). The last 2000 yr is based on archeomagnetic data from McElhinny and Senanayake (1982) and a modern moment of 8.0 Am². The error bars represent standard deviations.

Figure 3.8. Effects of paleointensity on TCN production rates at different latitudes and altitudes, using the Sint-200 relative paleointensity record (Fig. 3.7) to estimate dipole moment. The effect is expressed as the ratio of time-averaged production rate, P(t), to the present production rate at the geographic site latitudes (0° to 60°) at 2000 m elevation. Inset A shows the standard deviation in time-averaged production rates over the past 200 kyr, one way of showing that the greatest variability is felt at 20° latitude (negative latitude represents south). Inset B shows the effect of paleointensity variation at different elevations (sea level, 1, 2 and 3 km) at 20° latitude. All spatial scaling was done using Table 1 in Lal (1991).

Figure 3.9. Impact of a short-lived high amplitude intensity change on TCN production rates. The thick solid line shows the effect on production rates if the magnetic field was constant except during the Brunhes-Matuyama transition. The curve show the effect of decreasing a intensity to 20% of the modern intensity between 790 and 770 kyr. The effect reaches 3% for exposures that began within a few 100 kyr before the reversal. The upper dashed curve is the P(t)/P(today) based on Valet and Meynadier (1993) paleointensity for exposure times between 1000 and 600 kyr. For comparison, the thin dashed line is the same data, but relative to an average production rate (today’s magnetic field appears to be almost 1.5 times stronger than the average intensity over the past 1 Myr).

Figure 3.10 Comparison of the effects on geomagnetic latitude of variation in intensity (long dashes) and dipole axis position (thick solid curve). Upper curves for the Wind River Range, United States (geographic latitude 43°N). Lower curves for Southern Alps, New Zealand (geographic latitude 44°S). Paleointensity effects based on Sint-200 curve (Fig. 3.5.6-1). Dipole wobble effects modeled from VGPs of Ohno and Hamano (1992). Fine dashed line is geographic latitude.

Figure 3.11 Effects of variations in the geomagnetic field on TCN production rates, expressed as relative production rate. Dashed line is the change in time averaged production rate due to paleointensity variation according to the Sint-200 data (Fig. 3.7). Thin solid line is the effect on production rate due to secular variations in the dipole axis position. Thick line is the net geomagnetic field effect, calculated as described in the text. All graphs are for 1 km elevation for comparison. A. Wind River Range, Wyoming, USA, 43°N, 251°E. B. Echo Lake, Sierra Nevada, California, USA (39.7°N, 286°E). C. Hawaii, Pacific Ocean (21°N, 209°E). D. Prescott Island, Central Arctic, Canada (74°N, 263°E). E. Southern Alps, New Zealand (44°S, 169°E). F. Gobi Desert, China (41°N, 113°E).

Figure 3.12 Comparison of spallogenic production rates on a stationary surface in the central Arctic with those on surfaces that have been isostatically rebounding at an exponential rate (emergence rate from Dyke et al. (1991) based on 40 calibrated radiocarbon dated driftwood fragments on beaches around Prince of Wales island, with emergence history adjusted for eustatic sea level rise estimated from Fairbanks(1989)).

Figure 3.13. Scaling factor for variation of production rates with elevation and latitude (S_el), based on Lal (1991). Contours are elevation in km.

Figure 3.14 Illustration of geometrical quantities used in derivation of the shielding factor for sloping surfaces.

Figure 3.15. Total topographic scaling (S_T) as a function of surface dip angle and topographic shielding. Degree labels on curves refer to the angle of shielding by surrounding, axially symmetric, topographic features.

Figure 3.16 Effective attenuation length (L_f,e) as a function of surface dip angle and topographic shielding. Degree labels on curves refer to the angle of shielding by surrounding, radially symmetric, topographic features.

Figure 3.17 Effects of shielding by snow of common densities and thicknesses. Calculated for a spallogenic nuclide, assuming an otherwise simple exposure, with snow shielding instantaneously applied for 4 months each year. This is a multiplicative effect so the deviation can apply to any exposure age.

Figure 3.18a Variation of sample thickness scaling factor with sample thickness for spallation scaling factor (Q_s), thermal neutron scaling factor (Q_th), and epithermal scaling factor (Q_eth). Calculations are for an average low-Ca granite (Fabryka-Martin, 1988) with 1% volumetric water content and bulk density of 2.7 g cm^-2.

Figure 3.18b Variation in apparent age as a function of assumed rock density and sample thickness. Rock was a quartz arenite with actual density of 2.7 g cm^-3 and actual sample thickness of 5 cm.

Figure 3.19 Chlorine-36 concentration in Fabryka-Martin (1988) average low-Ca granite at sea level and high latitude as a function of surface exposure age and surface erosion rate.

Figure 3.20 Variation of apparent surface exposure age with erosion rate for different proportions of spallation and low-energy neutron production. The spallation-only curve would approximate the effects for ³He, ¹⁰Be, ¹⁴C, ²¹Ne, and ²⁶Al for shallow samples or low erosion rates where muonic contributions were not significant. Hypothetical sample was Fabryka-Martin (1988) average low-Ca granite at sea level and high latitude. Chlorine concentration and ³⁶Cl concentration were varied to maintain constant zero-erosion age while changing proportions of production reactions. 54% spallation production corresponds to 200 ppm Cl, the actual average value from Fabryka-Martin (1998).

Figure 3.21 Why use two isotopes? A single TCN measurement will at best provide a minimum estimate of an exposure age, because in most cases erosion reduces the concentration. After 30 kyr of exposure, in situ ¹⁴C at sea level high latitude (P_14C(Zo) = 20 atoms·g^-1·yr^-1) is within 97% of the secular equilibrium concentration (1.65 x 10⁵ atoms/g, at e = 0 mm/kyr). For higher erosion rates (e = 1, 3, and 10 mm/kyr) the ¹⁴C saturation concentration is lower (respectively illustrated as long dash, dash-dot, and dot) and equilibrium is attained earlier. On a surface known to predate the last glacial maximum, a measurement of ‘C’ = 1.59 x 10⁵ atoms/g would indicate that the erosion rate of the surface was 3 mm/kyr. Once the erosion rate is known it is possible to use a stable or longer-lived isotope (which has not reached saturation) to determine the actual exposure age. In this example, the measured concentration of ¹⁰Be (P_10Be(Zo) = 5.5 atoms·g^-1·yr^-1) is ‘Be’= 5.05 x 10⁵ atoms/g, which at e = 3 mm/kyr corresponds to an exposure age of 127 kyr (assuming ideal conditions as described in text).

Figure 3.22. TCN ratio diagram for ²⁶Al and ¹⁰Be. (a) Production curves for ²⁶Al (P_26Al(Zo) = 33.6 atoms·g^-1·yr^-1) and ¹⁰Be (P_10Be(Zo) = 5.5 atoms·g^-1·yr^-1), with production ratio of 6.1 (these production rates will vary with sites above sea level or l_m <60°). Additional curves for erosion rates of e = 1, 3, and 10 mm/kyr as in Fig. 3.21. The change in ²⁶Al/¹⁰Be also shown for e = 0, 1, 3, and 10 mm/kyr. (b) As in (a), ²⁶Al/¹⁰Be shown for e = 0, 1, 3, and 10 mm/kyr, but plotted (traditionally) against log ¹⁰Be concentration (normalized for production at sea level and high latitude). Samples with ratios plotting on the upper curve can be interpreted at have no erosion, and total exposure duration corresponds to distance along the e = 0 ratio curve. Samples plotting on or between erosion curves have experienced the modeled erosion rate (assumed continuous, constant, and gradual), and the total exposure time corresponds to the distance along the erosion trajectory.

Figure 3.23 TCN ratio diagrams. (a) ²⁶Al/¹⁰Be, as in Fig. 3.22b. Samples plotting below the steady state erosion island must have had experienced complicated exposure history (partial or complete shielding or plucking). The measured ratio could be explained by an infinite number of trajectories involving exposure, subaerial erosion, burial, or plucking. An example of complete shielding is given. If geological observations can decipher which surficial processes are likely, then the pathways illustrated can provide constraints on the minimum exposure age, minimum complete burial duration, and average plucking depth. Low ratios in allochthonous sediment also implies inheritance, so multiple nuclides are useful in depth profile studies. (b) ¹⁴C/¹⁰Be plot showing how the ratio behaves in the case of cliff retreat by block falls (i.e. erosion, not burial, caused the low ratios). If a cliff face was episodically retreating over the last 40 kyr, the ratios at any given time for the final surface are shown by the lowest curve (+). In this scenario, the cliff was exposed for 19 ka, then a 50 cm thick block fell, then the cliff was exposed for another 9 ka, then a 20 ka block fell, then another 10 ka, and finally another 50 cm block fell very recently (e.g. the block can be seen below the cliff, and the surface is very fresh). The average erosion rate is 120 cm over 38 kyr or e = 3.16 cm/kyr. In reality, we do not know this retreat history of the cliff. We can, however, provide an estimate of the minimum possible erosion rate to achieve the measured ratio. The dashed curves are calculated for the scenario that the cliff only underwent one block fall just before the sample was collected. The curves therefore provide the minimum possible erosion rate that could explain the measured ratio (considering 1s uncertainty = 8%, AMS and ICP.OES measurements only, see §6.1 text). In this case, the dashed curves bracket the minimum average erosion, e = 2.33 - 2.36 cm/kyr.

Figure 3.24. Ratio diagram, ³⁶Cl/¹⁰Be vs. ³⁶Cl (log concentration, normalized, as in previous figures) for boulder samples from moraines in the Wind River Range, Wyoming (from Phillips et al.,1997). Position of samples indicates generally very low erosion rates for boulder surfaces.

Figure 3.25 Depth profiles for spallogenic ¹⁰Be showing the model concentration for exposure scenarios described in the text.

Figure 3.26 Depth profiles of ³⁶Cl concentration measured at two locations close to the Socorro Canyon Fault scarp. The control profile was sampled topographically above the scarp, in a location with little erosion. Continuous curve illustrates calculated ³⁶Cl concentration after 140 ka of exposure. Star indicates average ³⁶Cl inheritance of clasts in deposit. Footwall profile was sampled 1 m above fault plane, in location of maximum erosion. The effect of the erosion is seen in the lower ³⁶Cl concentrations in this profile. The ³⁶Cl accumulation under the eroding surface, shown by the dashed curve, was modeled using a topographic diffusion-equation approach.

Figure 4.1. TCN precision based on multiple ages on a single landform. Filled circles are ¹⁰Be measurements on fresh granitic boulders with heights above ground > 1.2 m on a single moraine at 3200 m asl; coefficient of variation is 4.8% (Gosse et al., 1995a). Open circles are ¹⁰Be measurements on granodioritic boulders (H > 2 m) with varying degrees of weathering, on tightly nested recessional moraines of the Half Moon Lake Lobe near Pinedale, Wyoming, 2300 m asl; coefficient of variation is 4.6% (Gosse et al., 1995b). Measurements were done over two years with different but low chemical blanks (¹⁰Be/⁹Be averaged <2 x 10^-15), at the University of Pennsylvania.

Figure 4.2. Possible effects of lithology on exposure dating. Measurements of ¹⁰Be and ²⁶Al on 13 boulders (H > 2 m) on the crests of broad Bull Lake-age moraine ridges of the Fremont Lake Lobe, Wind River Range, Wyoming. Lithology numbers are ranked in order of increasing resistance to erosion and difficulty in sampling (1: weathered plagioclase porphyroblastic granodiorite; 2: plagioclase porphyroblastic granodiorite; 3: weathered granite; 4: fresh, unweathered granite). These Bull Lake moraines are believed to approximately coincide with Oxygen Isotope Stage 6 glaciation. The correlations are weak, but the positive trends in the ²⁶Al and ¹⁰Be ages indicate that the more resistant lithologies tend to erode less. The measured ²⁶Al/¹⁰Be also decrease with decreasing resistance to erosion. With the exception of inheritance, other geological factors that may cause variability should be constant. Although samples were collected from different recessional moraines of the same glaciation, there is no correlation between age and stratigraphic position.