3.2 Radioisotope 14C

The most commonly used cosmogenic isotope is radiocarbon 14C. This radionuclide is an unstable isotope of carbon with a half-life ( ) T1∕2 of about 5730 years. Since the radiocarbon method is extensively used in other science disciplines where accurate dating is a key issue (e.g., archeology, paleoclimatology, quaternary geology), it was developed primarily for this task. The solar-activity–reconstruction method, based on radiocarbon data, was initially developed as a by-product of the dating techniques used in archeology and Quaternary geology, in an effort to improve the quality of the dating by means of better information on the 14C variable source function. The present-day radiocarbon calibration curve, based on a dendrochronological scale, uninterruptedly covers the whole Holocene (and extending to 26,000 BP) and provides a solid quantitative basis for studying solar activity variations on the multi-millennial time scale.

3.2.1 Measurements

Radiocarbon is usually measured in tree rings, which allows an absolute dating of the samples by means of dendrochronology. Using a complicated technique, the 14C activity4 A is measured in an independently dated sample, which is then corrected for age as

( ) ∗ 0.693 t A = A ⋅ exp ------- , (5 ) T1∕2
where t and T1∕2 are the age of the sample and the half-life of the isotope, respectively. Then the relative deviation from the standard activity Ao of oxalic acid (the National Bureau of Standards) is calculated:
( A ∗ − Ao ) δ14C = -------- ⋅ 1000. (6 ) Ao
After correction for the carbon isotope fractionating (account for the 13C isotope) of the sample, the radiocarbon value of Δ14C is calculated (see details in Stuiver and Pollach, 1977).
Δ14C = δ14C − (2 ⋅ δ13C + 50 ) ⋅ (1 + δ14C∕1000 ), (7 )
where δ13C is the per mille deviation of the 13C content in the sample from that in the standard belemnite sample calculated similarly to Equation 6View Equation. The value of Δ14C (measured in per mille ‰) is further used as the index of radiocarbon relative activity. The series of Δ14C for the Holocene is presented in Figure 5View ImageA as published by the INTCAL04 (which is an update of the INTCAL98 series – Stuiver et al., 1998Jump To The Next Citation Point) collaboration of 21 dating laboratories as a result of systematic precise measurements of dated samples from around the world (Reimer et al., 2004Jump To The Next Citation Point). The most recent INTCAL09 data-set is available at External Linkhttp://www.radiocarbon.org/IntCal09.htm.UpdateJump To The Next Update Information
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Figure 5: Radiocarbon series for the Holocene. Upper panel: Measured content of Δ14C in tree rings by INTCAL-98/04 collaboration (Stuiver et al., 1998Jump To The Next Citation PointReimer et al., 2004Jump To The Next Citation Point). The long-term trend is caused by the geomagnetic field variations and the slow response of the oceans. Lower panel: Production rate of 14C in the atmosphere, reconstructed from the measured Δ14C (Usoskin and Kromer, 2005Jump To The Next Citation Point).

A potentially interesting approach has been made by Lal et al. (2005), who measured the amount of 14C directly produced by CR in polar ice. Although this method is free of the carbon-cycle influence, the first results, while being in general agreement with other methods, are not precise.

3.2.2 Production

The main source of radioisotope 14C (except anthropogenic sources during the last decades) is cosmic rays in the atmosphere. It is produced as a result of the capture of a thermal neutron by atmospheric nitrogen

14N + n → 14C + p. (8 )
Neutrons are always present in the atmosphere as a product of the cosmic-ray–induced cascade (see Section 3.1.3) but their flux varies in time along with the modulation of cosmic-ray flux. This provides continuous source of the isotope in the atmosphere, while the sinks are isotope decay and transport into other reservoirs as described below (the carbon cycle).

The connection between the cosmogenic-isotope–production rate, Q, at a given location (quantified via the geomagnetic latitude λG) and the cosmic-ray flux is given by

∫ ∞ Q = S(P, ϕ) Y(P ) dP , (9 ) Pc(λG )
where Pc is the local cosmic-ray–rigidity cutoff (see Section 3.1.2), S (P,ϕ ) is the differential energy spectrum of CR (see Section 3.1.1) and Y (P ) is the differential yield function of cosmogenic isotope production (Castagnoli and Lal, 1980Jump To The Next Citation PointMasarik and Beer, 1999Jump To The Next Citation Point2009Jump To The Next Citation Point).UpdateJump To The Next Update Information Because of the global nature of the carbon cycle and its long attenuation time, the radiocarbon is globally mixed before the final deposition, and Equation 9View Equation should be integrated over the globe. The yield function Y (P ) of the 14C production is shown in Figure 6View ImageA together with those for 10Be (see Section 3.3.2) and for a ground-based neutron monitor (NM), which is the main instrument for studying cosmic-ray variability during the modern epoch. One can see that the yield function increases with the energy of CR. On the other hand, the energy spectrum of CR decreases with energy. Accordingly, the differential production rate (i.e., the product of the yield function and the spectrum, F = Y ⋅ S – the integrand of Equation 9View Equation), shown in Figure 6View ImageB, is more informative. The differential production rate reflects the sensitivity to cosmic rays, and the total production rate is simply an integral of F over energy above the geomagnetic threshold. One can see that the sensitivity of 14C to CR peaks at a few GeV and is quite close to that of a neutron monitor (Beer, 2000Jump To The Next Citation Point).
View Image

Figure 6: Differential production rate for cosmogenic isotopes and ground-based neutron monitors as a function of cosmic-ray energy. Panel A): Yield functions of the globally averaged and polar 10Be production (Webber and Higbie, 2003Jump To The Next Citation Point), global 14C production (Castagnoli and Lal, 1980Jump To The Next Citation Point), polar neutron monitor (Clem and Dorman, 2000) as well as the energy spectrum of galactic cosmic protons for medium modulation (ϕ = 550 MV). Panel B): The differential production rate for global and polar 10Be production, global 14C production, and polar neutron monitor.

Thanks to the development of atmospheric cascade models (Section 3.1.3), there are numerical models that allow one to compute the radiocarbon production rate as a function of the modulation potential ϕ and the geomagnetic dipole moment M. The overall production of 14C is shown in Figure 7View Image.

The production rate of radiocarbon, Q14C, can vary as affected by different factors (see, e.g., Damon and Sonett, 1991Jump To The Next Citation Point):

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Figure 7: Globally-averaged production rate of 14C as a function of the modulation potential ϕ and geomagnetic dipole moment M, computed using the yield function by Castagnoli and Lal (1980), LIS by Burger et al. (2000) and cosmic-ray–modulation model by Usoskin et al. (2005aJump To The Next Citation Point). Another often used model (Masarik and Beer, 1999Jump To The Next Citation Point) yields a similar result.

Therefore, the production rate of 14C in the atmosphere can be modelled for a given time (namely, the modulation potential and geomagnetic dipole moment) and location. The global production rate Q is then obtained as a result of global averaging.

There is still a small unresolved discrepancy in the absolute value of the modeled 14C production rate. Different models yield the global-average production rate of 1.75 – 2.3 atoms cm–2 s–1 (see, e.g., O’Brien, 1979Jump To The Next Citation PointMasarik and Beer, 1999Jump To The Next Citation PointGoslar, 2001Jump To The Next Citation PointUsoskin et al., 2006bJump To The Next Citation Point, and references therein), which is consistent with a direct estimate of the radiocarbon reservoir, based on analyses of the specific 14C activity on the ground, 1.76 – 2.0 (Suess, 1965Damon et al., 1978Jump To The Next Citation PointO’Brien, 1979Jump To The Next Citation Point). On the other hand, the steady-state production calculated from the 14C inventory in the carbon-cycle model (see Section 3.2.3) typically yields 1.6 – 1.7 atoms cm–2 s–1 for the pre-industrial period (e.g., Goslar, 2001, and references therein). Thus, results obtained from the carbon cycle models and production models agree only marginally in the absolute values, and this needs further detailed studies. In 14C-based reconstructions, the pre-industrial steady-state production is commonly used.

3.2.3 Transport and deposition

Upon production cosmogenic radiocarbon gets quickly oxidized to carbon dioxide CO2 and takes part in the regular carbon cycle of interrelated systems: atmosphere-biosphere-ocean (Figure 4View Image). Because of the long residence time, radiocarbon becomes globally mixed in the atmosphere and involved in an exchange with the ocean. It is common to distinguish between an upper layer of the ocean, which can directly exchange CO2 with the air and deeper layers. The measured Δ14C comes from the biosphere (trees), which receives radiocarbon from the atmosphere. Therefore, the processes involved in the carbon cycle are quite complicated. The carbon cycle is usually described using a box model (Oeschger et al., 1974Jump To The Next Citation PointSiegenthaler et al., 1980Jump To The Next Citation Point), where it is represented by fluxes between different carbon reservoirs and mixing within the ocean reservoir(s), as shown in Figure 8View Image. Production and radioactive decay are also included in box models. Free parameters in a typical box model are the 14C production rate Q, the air-sea exchange rate (expressed as turnover rate κ), and the vertical–eddy-diffusion coefficient K, which quantifies ocean ventilation. Starting from the original representation (Oeschger et al., 1974), a variety of box models have been developed, which take into account subdivisions of the ocean reservoir and direct exchange between the deep ocean and the atmosphere at high latitudes. More complex models, including a diffusive approach, are able to simulate more realistic scenarios, but they require knowledge of a large number of model parameters. These parameters can be evaluated for the present time using the bomb test – studying the transport and distribution of the radiocarbon produced during the atmospheric nuclear tests. However, for long-term studies, only the production rate is considered variable, while the gas-exchange rate and ocean mixing are kept constant. Under such assumptions, there is no sense in subdividing reservoirs or processes, and a simple carbon box model is sufficient.

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Figure 8: A 12-box model of the carbon cycle (Broeker and Peng, 1986Siegenthaler et al., 1980).The number on each individual box is the steady-state Δ14C of this particular reservoir expressed in per mil. (After Bard et al., 1997Jump To The Next Citation Point)

Using the carbon cycle model and assuming that all its parameters are constant in time, one can evaluate the production rate Q from the measured Δ14C data. This assumption is well validated for the the Holocene (Damon et al., 1978Jump To The Next Citation PointStuiver et al., 1991Jump To The Next Citation Point) as there is no evidence of considerable oceanic change or other natural variability of the carbon cycle (Gerber et al., 2002), and accordingly all variations of Δ14C predominantly reflect the production rate. This is supported by the strong similarity of the fluctuations of the 10Be data in polar ice cores (Section 3.3) compared to 14C, despite their completely different geochemical fate (Bard et al., 1997Jump To The Next Citation Point). However, the changes in the carbon cycle during the last glaciation and deglaciation were dramatic, especially regarding ocean ventilation; this and the lack of independent information about the carbon cycle parameters, make it hardly possible to qualitatively estimate solar activity from 14C before the Holocene.

First attempts to extract information on production-rate variations from measured Δ14C were based on simple frequency separations of the signals. All slow changes were ascribed to climatic and geomagnetic variations, while short-term fluctuations were believed to be of solar origin. This was done by removing the long-term trend from the Δ14C series and claiming the residual as being a series of solar variability (e.g., Peristykh and Damon, 2003Jump To The Next Citation Point). This oversimplified approach was natural at earlier times, before the development of carbon cycle models, but later it was replaced by the inversion of the carbon cycle (i.e., the reconstruction of the production rate from the measured 14C concentration). Although mathematically this problem can be solved correctly as a system of linear differential equations, the presence of fluctuating noise with large magnitude makes it not straightforward, since the time derivative cannot be reliably identified. One traditional approach (e.g., Stuiver and Quay, 1980Jump To The Next Citation Point) is based on an iterative procedure, first assuming a constant production rate, and then fitting the calculated Δ14C variations to the actual measurements using a feedback scheme. A concurrent approach based on the presentation of the carbon cycle as a Fourier filter (Usoskin and Kromer, 2005) produces similar results. The production rate Q14C for the Holocene is shown in Figure 5View Image and depicts both short-term fluctuations as well as slower variations, mostly due to geomagnetic field changes (see Section 3.2.5).

3.2.4 The Suess effect and nuclear bomb tests

Unfortunately, cosmogenic 14C data cannot be easily used for the last century, primarily because of the extensive burning of fossil fuels. Since fossil fuels do not contain 14C, the produced CO2 dilutes the atmospheric 14CO2 concentration with respect to the pre-industrial epoch. Therefore, the measured Δ14C cannot be straightforwardly translated into the production rate Q after the late 19th century, and a special correction for fossil fuel burning is needed. This effect, known as the Suess effect (e.g., Suess, 1955) can be up to − 25‰ in Δ14C in 1950 (Tans et al., 1979), which is an order of magnitude larger than the amplitude of the 11-year cycle of a few per mil. Moreover, while the cosmogenic production of 14C is roughly homogeneous over the globe and time, the use of fossil fuels is highly nonuniform (e.g., de Jong and Mook, 1982) both spatially (developed countries, in the northern hemisphere) and temporarily (World Wars, Great Depression, industrialization, etc.). This makes it very difficult to perform an absolute normalization of the radiocarbon production to the direct measurements. Sophisticated numerical models (e.g., Sabine et al., 2004Mikaloff Fletcher et al., 2006) aim to account for the Suess effect and make good progress. However, the results obtained indicate that the determination of the Suess effect does not yet reach the accuracy required for the precise modelling and reconstruction of the 14C production for the industrial epoch. As noted by Matsumoto et al. (2004), “…Not all is well with the current generation of ocean carbon cycle models. At the same time, this highlights the danger in simply using the available models to represent state-of-the-art modeling without considering the credibility of each model.” Note that the atmospheric concentration of another carbon isotope 13C is partly affected by land use, which has also been modified during the last century.

Another anthropogenic activity greatly disturbing the natural variability of 14C is related to the atmospheric nuclear bomb tests actively performed in the 1960s. For example, the radiocarbon concentration nearly doubled in the early 1960s in the northern hemisphere after nuclear tests performed by the USSR and the USA in 1961 (Damon et al., 1978). On one hand, such sources of momentary spot injections of radioactive tracers (including 14C) provide a good opportunity to verify and calibrate the exchange parameters for different carbon -cycle reservoirs and circulation models (e.g., Bard et al., 1987Sweeney et al., 2007). Thus, the present-day carbon cycle is more-or-less known. On the other hand, the extensive additional production of isotopes during nuclear tests makes it hardly possible to use the 14C as a proxy for solar activity after the 1950s (Joos, 1994).

These anthropogenic effects do not allow one to make a straightforward link between pre-industrial data and direct experiments performed during more recent decades. Therefore, the question of the absolute normalization of 14C model is still open (see, e.g., the discussion in Solanki et al., 2004Jump To The Next Citation Point2005Jump To The Next Citation PointMuscheler et al., 2005Jump To The Next Citation Point).

3.2.5 The effect of the geomagnetic field

As discussed in Section 3.1.2, knowledge of geomagnetic shielding is an important aspect of the cosmogenic isotope method. Since radiocarbon is globally mixed in the atmosphere before deposition, its production is affected by changes in the geomagnetic dipole moment M, while magnetic-axis migration plays hardly any role in 14C data.

The crucial role of paleomagnetic reconstructions has long been known (e.g., Elsasser et al., 1956Kigoshi and Hasegawa, 1966). Many earlier corrections for possible geomagnetic-field changes were performed by detrending the measured Δ14C abundance or production rate Q (Stuiver and Quay, 1980Jump To The Next Citation PointVoss et al., 1996Jump To The Next Citation PointPeristykh and Damon, 2003Jump To The Next Citation Point), under the assumption that geomagnetic and solar signals can be disentangled in the frequency domain. Accordingly, the temporal series of either measured Δ14C or its production rate Q is decomposed into the slow changing trend and faster oscillations. The trend is supposed to be entirely due to geomagnetic changes, while the oscillations are ascribed to solar variability. Such a method, however, obliterates all information on possible long-term variations of solar activity. Simplified empirical correction factors were also often used (e.g., Stuiver and Quay, 1980Jump To The Next Citation PointStuiver et al., 1991Jump To The Next Citation Point). The modern approach is based on a physics-based model (e.g., Solanki et al., 2004Jump To The Next Citation PointVonmoos et al., 2006Jump To The Next Citation Point) and allows the quantitative reconstruction of solar activity, explicitly using independent reconstructions of the geomagnetic field. In this case the major source of errors in solar activity reconstructions is related to uncertainties in the paleomagnetic data (Snowball and Muscheler, 2007). These errors are insignificant for the last millennium (Usoskin et al., 2006aJump To The Next Citation Point), but become increasingly important for earlier times.

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