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     Last revised: Feb 18th 2001

            A chemical element is identified by its Atomic Number, Z, which defines both the number of protons (Proton Number) in the nucleus and the number of electrons in the neutral atom. The number of neutrons the atom contains is referred to as the Neutron Number, and the sum of the Proton and Neutron Numbers is the Atomic Mass Number. In the following notation for iron:


the value 57 refers to the atomic mass number of iron, and the value 26 to the atomic or proton number. Atoms that can be uniquely identified in terms of their proton numbers and atomic mass numbers are collectively called nuclides. Nuclides that have the same atomic number but different neutron numbers are called isotopes of an element. Nuclides with the same neutron number but different mass number are called isotones, whereas those with different proton or neutron numbers and the same mass number are called isobars. There are 81 stable elements comprising 264 stable nuclides.
         Unstable nuclides are calle radionuclides. Neutron rich radionuclides decay by ejecting an electron when an neutron is converted to a proton, e.g. 87Rb (Z=37) to 87Sr (Z=38), whereas proton-rich radionuclides tend to convert to isotopes of lower atomic number by capturing an electron, e.g. 40K (Z=19) to 40Ar (Z=18). These reactions are isobaric. Decay may also take place by the emission of a Helium atom or alpha-particle composed of 2 protons and 2 neutrons, e.g. 147Sm to 143Nd, whereby the atomic mass number decreases by 4 mass units, and the proton and neutron numbers by 2 units.
 Almost all of the elements exist in several isotopic forms, but only about 250 elements are stable and exist in measurable quantities, and only 51 of the known radioactive isotopes exist in nature.

        Elements of atomic mass number less than 20 (e.g. H, O, N, S, C) exhibit variations in isotopic composition that are detectable by modern means of measurement, whereas most elements with mass number greater than 20 tend to show constant isotopic composition. The exceptions, K, Ar, Rb, Sr, Sm, Nd, Rh, Os, and U, Th, and Pb, vary in composition because they include varying proportions of the parent and daughter products of radioactive disintegration.
        The measurement of the relative concentration of isotopes in rocks is extremely useful as a means:
            1) to establish the tectonic environment of formation of rocks;
            2) to determine the age of rocks;
            3) to estimate the physical conditions of the hydrosphere and atmosphere at the time of
sedimentary rock deposition; i.e. to act as chemostratigraphic markers.
            4) to estimate the nature of the source rocks of hydrothermal mineral deposits.

    References:     Faure, G., Principles and Applications of Geochemistry, 2nd ed, Prentice Hall.
                            Brownlow, A.H., Geochemistry, 2nd ed, Prentice Hall.
                            Allegre, C-J., and Michard, G., Introduction to Geochemistry, Reidel.
                            Isotopic studies in Paleontology, Journal of the Geological Society, v. 154, p. 293-356

    The Stable Light Isotopes

        The non-radioactive light elements oxygen, carbon and sulphur, which are commonly referred to as Stable Isotopes, are valuable in understanding geological processes because they may undergo fractionation when the phase in which they occur changes state (e.g. oxygen in water changing from liquid to gas or solid), or participates in a chemical reaction involving a change of state (e.g. sulphate versus sulphide in the case of sulphur). Consequently, the oxygen isotopic composition of sea water differs from that of both polar ice, atmospheric water vapour, or rain water, where the seas and polar ice can be considered to represent independent end-member water reservoirs in quasi-equilibrium with each other. The Raleigh fractionation of the oxygen isotopes takes place during the sequential removal of rain water from  atmospheric vapour as the vapour is transferred from its place of origin (evaporation) in the tropics towards the polar ice reservoir. Note that if there were no polar ice-reservoir most of the atmospheric water vapour would be returned to the oceans, and all things else being equal, the oxygen isotopic composition of the oceans would be constant.
         Similarly, carbon in sea water is fractionated as a result of biogenic activity, carbon-based life forms favouring the concentration of 12C over 13C. However, as in the case of water vapour, once the 'bugs' die their carbon would be returned to the oceans, which would therefore exhibit only a secular variation in isotopic composition related to variations in the rate of biogenic activity.  On the other hand, if a significant fraction of the 'deceased' were to be buried during sedimentation, the sedimentary rocks would constitute a separate carbon reservoir, and the isotopic composition of sea water would then reflect the relative rates of biogenic activity, sedimentation, and the erosion of carbon rock reservoirs. Fractionation would be a maximum at high rates of biogenic activity and sedimentation, and low rates of erosion.
        In the case of sulphur, the isotopic fractionation takes place as a result of the bacteria-mediated progressive (Raleigh) conversion of ocean water sulphate to 32S enriched hydrogen sulphide, and the removal of the latter into a separate rock reservoir as pyrite. Note again that if the hydrogen sulphide is not separated out of solution, or if all the sulphate were to converted to sulphide, the isotopic composition of the sea water would not change.


        Average abundance ratio 18O/16O ratio = 0.2/99.762 = 0.0020048 = 1/498.81

        Isotopes of oxygen are fractionated when they pass from the hydrosphere to the atmosphere such that, for example, oxygen becomes enriched in the light oxygen isotope 16O relative to the heavy oxygen isotope 18O. Numerically, the variation (known as d18O) is expressed in terms of the deviation of the ratio of 18O to 16O in the sample of material being analyzed from the mean value of 18O/16O in seawater (SMOW, standing for Standard Mean Ocean Water), viz,

            d18O = (18O/16Osample - 18O/16OSMOW)/18O/16OSMOW * 10^3 °/°°

(Note 1: 18O/16O ratio of SMOW is 0.0020052; if the sample is enriched in light oxygen 16O such that 18O/16O is less than this value, the d18O value will be negative )
(Note 2: the oxygen isotope composition of carbonates is commonly expressed relative to a carbonate standard known as PDB (Peedee belemnite), where d18O (SMOW) = 1.03086 d18O (PDB) + 30.86.; the d18O value of carbonates is usually much greater than that of SMOW).
(Note 3: the variation in relative concentration of Hydrogen and Deuterium tends to mimic that of 18O and 16O, repectively, and the variation relative to SMOW in this case is represented as dD.
        Mantle materials have d18O values of about +5 to +6, whereas rain water may have values as low as -50 at the South Pole. One might anticipate that atmospheric oxygen (O2) would also be relatively light. However, it is highly positive as a result of fractionation related to biological respiration (Dole effect) - biogenic materials always tend to concentrate the light isotope. Note that the isotopic composition of oxygen produced by photosynthesis is related to the isotopic composition of the hydrous enviroment.

        Oxygen isotope variation in rocks.

        Because rain water fractionates heavy oxygen out of atmospheric moisture (16O increases and 18O/16O decreases), the moisture therefore becomes progressively enriched in light oxygen as it moves from warm equatorial latitudes towards the cold polar regions. Polar ice therefore constitutes a reservoir of 16O enriched water separate from sea water. The isotopic composition of the latter is therefore controlled by the magnitude of the polar ice regions. Melting the polar ice caps would decrease the d18O value of sea water.

        Oxygen isotope variation in precipitated waters.

        Clay minerals formed during weathering may have positive d18O values, varying from +10 at high latitudes to +30 at low latitudes (nearer the equator), and carbonates here may have values as high as 34. (see Faure p. 353 for the values for marine shales.)
       The measurement of d18O values provides a relatively simple means of 'fingerprinting' fluids involved in hydrothermal mineralization, hot spring formation, and diagenesis. Note that in the case of hot springs (see following diagram) ground water oxygen 'gets heavier' (d18O is less negative) as it equilibriates with the rock through which it passes
        Oxygen isotope variation in thermal waters.

        In a less simple manner, the penetration of sea water through oceanic basalt causes the d18O of the basalt to initially increase and then to decrease once the temperature reaches that of the upper greenschist facies.
        While the removal of light water to form polar ice causes the d18O of sea water to increase,  the precipitation of  carbonate with a positive d18O value causes a decrease in the d18O value of seawater. Furthermore, a decrease in sea water temperature causes an increase in d18O of precipitated carbonate, thereby tending to accentuate the decrease in the d18O value of seawater. With a fall in temperature therefore, carbonate precipitation will tend to counterbalance any increase of the d18O of the sea water through the removal of water in the form of polar ice.On the other hand a decrease in average oceanic temperature will increase the solubility of carbonate, thereby minimizing the effect of carbonate precipitation. To relativize the effect of polar ice and carbonate precipitation,  it is necessary therefore to analyze the shells of both surface living planktonic foraminifera and the shells of deep-dwelling benthic species living at relatively constant temperature.

        Advanced Reading:  Huber, BT., MacLeod, K.G., and Wing, S.L., 1999. Warm Climates in Earth History. Cambridge University Press, 462p., US$115; review by Bednarski, J.M., Geoscience Canada, v. 27, 4, 189-191.


        Recommended background reading: Berner, R.A., 1999, A new look at the long-term carbon cycle. GSA Today, v. 9, no. 11, p. 1-6.

        Average 13C/12C ratio = 1.1/98.9 = 0.0111223 = 1/89.9091
        In calculating the d13C values of organic material, the carbon standard used is the limestone of a fossil belemnite from the Pee Dee Formation (PDB) of North Carolina.
        Living organisms preferentially concentrates 12C relative to 13C, and the d13C (PDB) isotope values of plants that use the C3 metabolic process ranges from -23 to -34 per mil, whereas those (corn, tropical plants) that use the C4 process ranges from -6 to -23 (Beerling, 1997, p.303). (The advent of C4 vegetation took place in the Miocene.) CO2 removed from soils by plant respiration is also enriched in 12C and therefore isotopically light (c. -27 per mil) in comparison with normal atmospheric CO2 (-6 per mil). Because of the preferential removal of 12C by plant respiration, palaeosol carbonates (carbonates in soils) will therefore evolve in the direction of higher (less negative) d13C values. On the other hand, fossil fuels are strongly enriched in 12C relative to 13C , that is they have lower or more negative d13C values reflective of  their organic derivation.
        Bicarbonate formed from atmospheric CO2 (d13C - -6 per mil. PDB) entering solution:
                                                                CO +  H2O = HCO3- +H +
is enriched in 13C, and when CaCO3 precipitates from bicarbonate solution:
                                                                Ca(HCO3)2  =  CaCO +  CO2  +  H2O
it is further enriched in this isotope. The d13C (PDB) of Phanerozoic marine carbonate rocks therefore tends towards a value of 0.
        [Note that precipitation of CaCO3 and release of CO2 to the atmosphere is favoured by rising temperatures. The increase in CO2 can only be mitigated by an increase in weathering reactions, e.g.
                              Primary material                                          Weathered material
                CaAl2Si2O8 (Anorthite) + 3H2O + 2CO2    =   Al2Si2O5(OH)4 (Clay) + Ca(HCO3)2]

        Secular changes in the isotopic composition of the aquatic primary producers are related to the concentration of CO2 in the atmosphere, where the isotope fractionation associated with the photosynthetic fixation of carbon is proportional to the log value of the concentration of dissolved CO2. The d13C of organic carbon can therefore be used to provide a likely record of the abundance of atmospheric CO2 (Beerling, p. 304), and from the limestone organic carbon record it would seem that atmospheric concentration of CO2 was high at the end of the Cretaceous (1000 ppm) and has decreased (< 400 ppm) ever since that time (Beerling p. 305). In contrast the 87Sr/86Sr of sea water has consistently increased since the end of the Cretaceous, implying that weathering/erosion of continental crust has been increasingly more effective.
        The value of 13C/12C relative to  that of a standard limestone (Peedee belemnite), d13C, is controlled 1) by the level of biotic activity in the oceans (which is itself controlled by the availability of nutrients); and 2) the rate of removal of the organic carbon (and of nutrients) by burial relative to its return to the oceans by weathering, solution, and the precipitation of carbonates. High rates of biotic activity and of carbon burial, that is high rates of removal of 12C, will cause sea water to develop a relatively high d13C value (e.g. +8 per mil). For this reason a d13C minimum (more 12C = decrease in the ratio 13C/12C  = negative d13C excursion) follows some sudden extinction events because the reduced biotic activity diminishes removal of 12C, whereas a relatively large amount of 12C represented by the deceased bugs is dissolved back into the seawater.
        High d13C values may therefore reflect high levels of biotic activity in well developed optimum-sized shelf seas receiving a strong flux of dissolved nutrients (phosphorus), as well as an abundant supply of clastic sediment to rapidly bury the biogenic carbon being produced. Polar ice caps would generate active deep-water cold water currents that would also be effective in supplying nutrients to the shelf, and the magnitude of the ice caps would control the degree of transgression or regression of the seas onto the continents.  Should conditions move towards a 'snowball' Earth, the shelves and associated biotic activity might be forced into a sharp recession, and d13C values would markedly decrease.  On the other hand, an increase in oceanic tectonic activity might cause excessive flooding of the continents (high flux of greenhouse CO2 leading to the melting of the ice caps), which in turn would turn off the supply of nutrients and of sediment. The flux of mantle derived CO2 with d13C of about -7 /mil would also tend to decrease the d13C of ocean water.

        Examples in the Geological Record
        The Paleo-Proterozoic included a period of negative d13C excursions perhaps coincident with the Huronian Gowganda glaciation, followed by three positive excursions in the interval 2.43 to 1.93 Ga (Geology 1998, p. 875). In contrast, d13C for the Meso-Proterozic until 1 Ga ago remained relatively constant with values of about 0. During the Neo-Proterozoic, d13C values once again increased to values greater than 10 (significant increase in biotic activity) punctuated by high-amplitude negative excursions coincident with periods of 'snowball' Earth glaciations; implying very sudden terminations of biotic activity.

        Secular variation in d13C since the Archean

        The late Ordovician (Hirnantian) faunal extinction was accompanied by the onset in growth of ice caps, a fall in sea level of a 100 metres, and an increase of d13C from 0 to +6 (all values per mil PDB) of sea water (there might be an increase in burial rates and decrease in erosion and therefore of the rate of transfer of 12C to the oceans, as well as an overall increase in biotic activity due to increased nutrient supply from the polar regions to the shelves), as well as of d18O from -4.5 to -1.5 (an excursion to less -ve values reflects the transfer of more light 16O to the icecaps).
Stable isotope curves for brachiopod compositions variation during the late Orodovician Hirnantian stage (Brenchley et al. 1997) - hirnantian1.jpg
Stratigraphic profile of  Upper Ordovician rocks of the Oslo (Norway) region (Brenchley et al, 1997) - hirnantian2.jpg
        Similar positive excursions (-1 to 7.5 d 13C), have been recognized at three levels in the Silurian (BGSA 1999, p. 1499) and correlated with sea-level lowstands, and the Carboniferous glaciation was marked by a positive d13C excursion to +6 per mil.
        A decrease in d13C of 3 per mil during the Artinskian and Kungarian of the early Permian, and a general decrease (more negative) change at the Permian - Triassic boundary (250 Ma), was commensurate with a relative decrease in 13C in the oceans and atmosphere, that is, an increase in 12C. By 250 Ma the supercontinent Pangaea was complete, but subject to uplift around its margins due to the tectonic compressive stress of collision. As a result, the vast peat reserves in the coal-bearing foreland basins of Pangaea were oxidized leading to a massive flux of CO2 with low d13C values into the oceans; plunging d13C values once again to about 0 per mil.
        At the Cretaceous-Tertiary boundary, reduced light levels, supposedly due to meteorite impact, inhibited phytoplankton, and then the zooplankton feeding on the phytoplankton, and other life forms feeding on the phytoplankton, causing a negative 13C excursion - that is more 12C was returned to the oceans than was removed by biotic activity.
        At the Cenomanian - Turonian boundary large quantities of organic carbon were produced causing a marked +ve d13C excursion for sea water. The increase in biotic activity was however accompanied by an increased burial rate of the carbon, thus causing nutrients to be removed from the system (i.e. no food). This caused a decline in photic zone coccolith production and carbonate accumulation rates, which in turn starved the zoooplankton and caused their selective extinction. There was therefore a return to lower d 13C values (negative d13C excursion; more 12C was dissolved into the oceans than was removed by the biotic activity). Higher temperatures at this time also reduced coccolith (carbonate) productivity.

        Isotope curves through the Cenomanian-Turonian boundary.

        Rocks that contain buried carbon are characterized by a -ve d13C values, and the carbonate in quartz veins containing Au flushed from sedimentary rocks invariably exhibits -ve d13C values (e.g. -20 per mil).

                THE BOX MODEL - Carbon in the ocean.
        The ocean is divided into to parts, an upper zone that is warm and well mixed, and a deep zone that is cold and stratified. The two zones are easily differentiated by their content of carbon and particularly by that of 14C. Carbon is exchanged between the two reservoirs as dissolved forms and and as a flux of particles. Thus:
        B is the descending flux of solid particles in moles/year, that dissolve in the lower zone;
        Vd is the volume of the deep zone.
        W and W' are the volumes of sea water descending from the upper zone to the lower and ascending from the lower to the upper, respectively, in m^3 per year.
        Cs and Cd are the concentrations of Carbon dissolved in the upper and lower zones, respectively.
        Rs (.95) and Rd (.8) are the ratios of 14C/C in the upper and lower zones, respectively, relative to 14C/C in the atmosphere. l = radioactive decay constant of 14C.
        If the system is in a steady state, then the volume descending from the upper zone to the lower zone equals the volume of water ascending in the reverse direction:
                                                                        W = W'
        and at equilibrium there is an equal exchange of Carbon between the shallow and deep zones:
                        Total C in descending flux (W.Cs + B) = total C in ascending flux
                                                   (W.Cd) W.Cs + B = W.Cd, (and therefore Cd > Cs)

        In the case of 14C equilibrium:
                                            14C in the descending flux = 14C in ascending flux
                                                              (W.Cs + B).Rs = W.Cd.Rs
        and                                            W.Cd.Rs = W.Cd.Rd + Vd.Cd.Rd.l
        where Rd < Rs because of the loss by radioactive disintegation of 14C in the the total volume Vd.
                                                            WRd + Vd.Rd.l = W.Rs
                                                            W.(Rs - Rd) = Vd.Rd.l
                                                            W/Vd = Rd.l /(Rs - Rd)
        and residence time = (total carbon in deep zone/carbon descending /year from upper zone to lower zone,
                                                    = Vd.Cd/W.Cd = Vd./W = (Rs - Rd)/(Rs.l )
          If l = 1.25 x 10-4 yr-1), the residence time is about 100 years.
        Study of the variation of 14C in the worlds oceans has lead to the distinction of 6 large oceanic reservoirs.


        Average abundance 34S/ 32S = 4.21/95.02= 0.0443065 = 1/22.570071

        32S/34S in various rock types.

        Sulphur fractionation is evident in the relative variation of the sulphur isotopes 32S and 34S, usually expressed as a d value relative to a standard 32S/34S ratio of 22.225, the value of the Canyon Diablo Troilite meteorite and close to that of most mantle derived mafic rocks.
        (Note 4: in some conventions the ratio of isotopes are given as the ratio of the more abundant isotope to the less abundant isotope, whereas the calculation of d values uses the ratios of heavier to lighter isotopes. Because 32S is more abundant than 34S, sulphur isotope ratios in the above graph are presented as 32S/34S and have values greater than unity. Consequently, positive d34S values in the graph plot to the left of 0, and negative values to the right.)
        Sulphur isotopes are fractionated because the heavier isotope is preferentially taken up by the compound in which sulphur is most strongly bonded, sulphate as distinct from sulphide in the case of 34S. The d34S values of sulphates in sea and fresh water range from +4 to +30, with a value of +20 more typical of sea water, whereas the sulphides associated with mineral deposits range from +7 to -7 in igneous rocks and +50 to -45 in sedimentary rocks.
        In recent marine sediments it has been observed (Kaplan 1983) that bacterial reduction of sulphate causes the dissolved H2S in pore water in the sediment to become enriched in 32S and the sulphate to become enriched in 34S. If the hydrogen sulphide escapes out of the pore water, the dissolved sulphate and total sulphur in the downward circulating pore water will consequently decrease, and the dissolved sulphate will be progressively enriched in 34S (Raleigh distillation). If the pore water is returned to the oceans, the latter would become enriched in 34S.  H2S produced from this sulphate by bacterial reduction will then also be relatively enriched in 32S but progressively less than that formed earlier.  If the H2S is fixed as pyrite, the latter will take on the enriched 32S isotopic characteristics of the H2S (negative d values) from which it was formed. On the other hand pyrite formed directly from the 34S enriched sulphate will exhibit positive d values.
        The isotope evolution of S in the oceans is very similar to that of Sr. Both show high d values during the late Proterozoic - Cambrian and the Present, and minimum values during the Mesozoic. The isotopic character of the Sulphur contributed by the continents to the oceans will however depend upon the relative contributions of weathered sulphates (evaporites) and sulphides (igneous and sedimentary rocks). The isotopic character of sea water will also be controlled by the relative rate of formation of evaporite deposits, the effectiveness of bacterial activity in reducing sulphate taken from the oceans, the rate of introduction of cold oxygenated waters from the polar regions (sulphate production versus biotic activity), and the rate of removal of sulphide by reactive volcanic Fe.
        What would be the optimal conditions for high 32S/34S levels (low d values) in ocean water:
    1) high rate of formation of evaporite and high erosion rates of d-negative sulphidic sediments; 2) low volcanic activity = low Fe = low sulphide removal; 3) high oxygenated water = conversion of insoluble pyrite to soluble sulphate without sulphur fractionation; 4) low bacterial rate of reduction of sulphate if sulphate is returned to the oceans.

        Variation in Sulphur and Strontium isotopes in seawater since the late Proterozoic.

        C/S ratios

        The C/S ratio of normal marine sediments (deposited beneath oxygenated waters) is c. 2.0 in Late Paleozoic and younger rocks, but as low as 0.5 in older Paleozoic sediments. The difference is attributed to the advent of land plants in the Late Silurian and the fact that terrestrial organic matter is less easily metabolized (less labile) than marine organic matter. The presence of less reactive plant material in the sediment therefore increases its C/S ratio. However, some Cambrian sediments have ratios as high as 1.4, suggesting that values as low as 0.5 may reflect the increased presence of reactive Fe in the form of volcanic glass that would trap the bacterial formed hydrogen sulphide as pyrite. That is, the lower C/S ratio is due to increased sulphur content rather than decreased carbon content.

        The C-S-O cycle

        Bacteria mediated:
        2CH2O + H2SO4 = 2CO2 + H2S + 2H2O (removal of C and S from the oceans to the atmosphere)
        2CO2 +2H2O = 2CH2O + 2O2 (conversion of CO2 to O2, and addition of C to the oceans)
        H2S + 2O2 = H2SO4 (return of S and O to the oceans from the atmosphere).

        These three equations balance out, and are rate rather than thermodynamically controlled. The rate relationship can be perturbed by, for example, the addition of volcanic ferrous iron, or the rate of burial of CH2O, or erosion of sulphate deposits, or removal of CO2 as carbonate from the oceans.

        The Radioactive Elements

        Radioactive elements are useful for two reasons:
        1) they allow the determination of the age of rocks and minerals;
        2) they can be used to 'fingerprint' the primary source of rock and mineral material.

        The main isotopic systems are those of K and Ar, Rb and Sr, Sm and Nd, U or Th and Pb, and Rh and Os, where in each case the first element of the pair is the parent isotope and the second element the daughter isotope. In the case of the 87Rb-87Sr nuclide pair, 87Rb, whose proton number is 37 and whose neutron number is 50, changes to 87Sr with a proton number of 38 and a neutron number of 49. In this case the change involves the conversion of one neutron to one proton plus one Beta particle and an antineutrino. On the other hand 147Sm (proton number 62) changes to 143Nd (proton number 60) by the loss of a 4He atom or alpha particle (2 protons and 2 neutrons), whereas the conversion of U to Pb takes place via a series of spontaneous fission reactions.

        The Law of Radioactivity

        In the following description we will use the example of Rb/Sr, although the same calculation applies to any of the parent-daughter pairs mentioned above. The law of radioactivity states that the rate of decay of the parent element at any time during its decay is proportional to the number of atoms of the parent present at that time. A plot of 87Rb as ordinate against Time as abscissa will therefore define a curve which will be negative and whose slope will decrease with the passage of time. The equation of the curve will be: -dRb/dt = l.Rb (where l is the decay constant) and therefore:
                                                            -dRb/Rb = l.dt
    Integrating dRb and dt from time t0 to time tn (now):]
                -(lnRbtn-lnRbt0) = l d t and ln[Rbt0/Rbtn] = l.dt and Rbt0 / Rbtn = e^l.dt ;
    and                                                     Rbt0 = Rbtn e^l.dt
       This relationship allows us to calculate the 87Rb content of a rock or mineral at any time in the past from its present day value.

        87Rb versus time.

        Since the amount of daughter product, 87Sr = 87Rbt0 - 87Rbtn and the initial amount of 87Sr in the rock at time t0 = Sri then the total amount of 87Sr = 87SrT = 87Sri + (87Rbt0 - 87Rbtn) = 87Sri + (87Rbtn e^l.dt - 87Rbtn)
                                               = 87Sri + 87Rbtn (e^l.dt - 1)
        Since SrT and Rbtn can be measured, and delta is a constant, the age of the rock, delta t, can be calculated if we also  know the value of Sr0, the initial amount of 87Sr in the rock. This however we cannot know, and even if we were to construct a second equation using data from a second specimen with a different 87Rb and Total 87Sr, we still cannot assume that 87Sri will be same in both specimens. The problem can be solved, however, if we assume that the relative proportions of the various Sr isotopes in all coeval and consanguinous samples is constant. In this case the ratio of 87Sr to the stable Sr isotope 86Sr will be the same in all specimens, even if the actual amount of 87Sr is different. The above equation can then be converted to the form:
                        (87Sr/86Sr)T = (87Sr/86Sr)i + 87Rbtn/86Sr (e^l.dt - 1))
        Two or more samples of the rock with different (87Sr/86Sr)T and 87Rbtn/86Sr values would then allow the writing of a set of simultaneous equations from which delta t can be calculated.

        87Rb/86Sr versus 87Sr/86Sr.

        The above equation has the form of a straight line, Y = AX + B, where (87Sr/86Sr)i, commonly known as the 87Sr/86Sr initial ratio (Sri), is the intercept, and (e^l.dt - 1) the slope of the line. Consequently, the values of these parameters can be determined by graphing 87SrT/86Sr against 87Rbtn/86Sr. Such a graph (above) is known as a Nicolaysen graph, and is the preferred method of representing isotopic data in the calculation of rock and mineral ages. Both Rb and Sr behave as relatively incompatible elements in basaltic liquids. Consequently, radiogenic Sr in the basaltic crust will grow at a faster rate than the primary mantle (Bulk Earth), which will itself produce radiogenic Sr at a faster rate than the depleted mantle. If at any time subsequent to the formation of the basalt, any or all of these independent reservoirs again undergoes melting, the Sri of the melts will reflect that of the reservoir from which they are derived. Melts derived from depleted mantle will have lower Sri values than melts derived from undepleted mantle, which will have Sri values lower than melts derived from the basaltic crust.

        87Sr/86Sr versus time.

        It is important to note that the Sri values of the melts from the reservoirs is independent of the mineralogy of the reservoir or the amount of melt produced, because the Sr isotopes are not fractionated by physical processes involving solids and liquids in the crust or mantle. The Sri values are determined by the initial Rb contents of the reservoirs following melting, and the time since the melting event. Because MORB basalts have low Sri we know that the mantle from which oceanic basalts are derived is depleted in Rb as well as in the associated elements K and Ba. On the other hand WP basalts have relatively high Sri values, indicating that the mantle source of WPB's is relatively enriched in Rb. Because Rb is fractionated into the upper continental crust (following partial melting of the first formed basaltic crust), the crust has a high Sri, with the highest values appearing in the oldest continental rocks.
        Note that  we can calculate the 87Rbtn/86Sr of any rock or mineral in the time past from the relationship:

                                                   (87Rbtn/86Sr)t0 = (87Rbtn/86Sr)tn.e^l.dt

Model Ages

        If we can assume that a rock is derived from the primitive mantle (usually referred to as BABI, “basaltic achondrite best initial” (the time of formation of the Earth and other planetary objects) or BE “the BULK EARTH”), the age of the melting event can be calculated knowing the Sr/Sr and Rb/Sr characteristics of the rock and of BABI/BE by writing a set of simultaneous equations based on the relationship:
                                (87Sr/86Sr)Total = (87Sr/86Sr)initial + Rbtn/86Sr (e^l.dt - 1)
        If the basalt was derived from a mantle source that has already been melted and is therefore depleted in incompatible elements such as Rb, the same calculation can be carried out by substituting the values of 'Depleted Mantle or DM' for those of the Bulk Earth. The relevant values for the present-day Bulk Earth and Depleted mantle are:
                                                87Sr/86Sr            87Rb/86Sr
            Bulk Earth (BABI)         .7045                .0827
            Depleted Mantle (DM)  .7033958          .05541494

        As illustrated in the following diagram, solving for delta t in the simultaneous equations is equivalent to determining the point of backward intersection of the the sample and Bulk Earth Sr/ Sr growth curves in a plot of Sr/Sr against Time. The age at the point of intersection is known as a 'model age' and is considered to represent the maximum age the basaltic material can have based on the assumption of either a Bulk Earth or Depleted Mantle source. Note that the Model Age relative to the Depleted Mantle will be older than the model age relative to the Bulk Earth.

        87Sr/86Sr versus time - Model age.

        In the case of sediments or granite derived from sediments with a mixed source, the model age is the average age of the source material. In this case all that can be said is that the age of the sediment must be younger than its model age. The true age of the sample, determined by some other method e.g. U-Pb in zircon, could be considerably younger than its model age, in which case it would be necessary to contemplate an origin in terms of mixing mantle derived material with a crustal component with higher Rb/Sr and Sr/Sri values.

        87Sr/86Sr versus time - model age by mixing.


        206Pb, 207Pb and 208Pb are produced by the radioactive decay of 238U, 235U and 232Th, respectively. The only non-radiogenic isotope of lead is 204Pb, and the isotopic composition of lead is therefore usually expressed as 206Pb/204Pb, etc, e.g.
                                        (206Pb/204Pb)T = (206Pb/204Pb)i + (238U/204Pb)tn (e^l.dt - 1)
        The main advantage of using zircon to date rocks is that zircon can be assumed to contain no initial lead, and the age equation reduces firstly to :
                                        (206Pb/204Pb)T = 238U/204Pb (e^l.dt - 1)
    and then to                                 206Pb/238U = (e^l238.dt - 1)
    For the nuclide pair 235U-207Pb, the equation would be :
                                                       207Pb/235U = (e^l235.dt - 1)
    AND                             206Pb/238U = 207Pb/235U ((e^l238.dt - 1)/(e^l235.dt - 1))

        Zircon ages are usually calculated using what are called 'Concordia Diagrams' involving plots of 207Pb/235U against 206Pb/238U, and where the concordia line is respresented by the latter equation. Distance measured along the line from the origin is a measure of the age of the zircon, and if the zircons have not suffered chemical disturbance they should plot on the line. The age of the zircon is then said to be concordant. If the data points do not plot on the concordia line, they are said to be discordant. In this case related zircons may plot on a straight 'mixing' line, such that the upper intercept of the line with the concordia line would represent the primary age of the zircon and the lower intercept the age of some subsequent metamorphic effect.


        In the case of the siderophile parent-daughter pair 187Re-187Os (most of the Re - Os resides in the core), melting of the mantle causes Re to be largely partitioned into the basaltic melt but Os to be retained in the mantle. Following a mantle melting event therefore, Os will grow at a much faster rate in the crust than in the depleted mantle. 187Os/188Os of the chondritic mantle is about 0.13, whereas 187Os/188Os of the highly radiogenic crust is about 1.7.  It is therefore relatively easy using Os isotopes to differentiate melts derived from the depleted mantle from those derived from basaltic crust.
        Re-Os  isotope studies indicate that the upper part of the lithospheric mantle beneath continental southern Africa was depleted by melting during the early Archean (low 187Os/186Os initial) and that the diamond bearing South African kimberlites represent small melts of enriched mantle below the radiogenic Os depleted mantle; that the Sudbury Irruptive was formed by melting of continental material rather than the mantle, and that the 1 billion year old Keweenawan lavas of the Lake Superior region include a component of mantle material highly depleted during the Archean. (Example reading: Asmerom, Y. and Walker, R.J., 1998. Pb and Os isotopic constraints on the composition and rheology of the lower crust. Geology, 26, 4, 359-362.)


    The parent - daughter pair 147Sm - 143Nd can be treated in the same way as Rb - Sr. However because depleted mantle is light REE depleted (REE pattern has positive slope), and the Bulk Earth (source of WP or Plume basalts) light REE enriched (REE pattern negative slope), and given that Nd is lighter than Sm, melts derived from the Depleted Mantle (high Sm/Nd) will tend to exhibit time integrated 143Nd/144Nd values greater than melts derived from Bulk Earth compositions (low Sm/Nd).
        The (143Nd/144Nd)initial  values of melts derived from the mantle can be expressed in terms of the Nd/Nd value for the BULK EARTH in the same way that oxygen isotopes are expressed relative to seawater, viz, eNd = (Nd/Ndsample - Nd/NdCHUR)/Nd/NdCHUR * 10^4 °/°° where CHUR stands for Chondritic Undepleted Reservoir The eNd values of present-day MORB are about 8 - 12, indicating that the mantle source of MORB is depleted relative to the Bulk Earth (Chondrite). The eNd of a rock is calculated using the Nd/Nd values of  the sample relative to the Bulk Earth at the time of formation of the rock sample.

    Variation of Nd/Nd versus time compared with eNd versus time for the mantle and crust - epsilnd.jpg

        If a similar index is calculated for (Sr/Sr)initial, the data may be plotted on an eNd versus eSr diagram. On such a diagram melts derived from a depleted mantle will have positive eNd values and negative eSr values, and the melts will tend to lie along a line with negative slope passing through the value for the Bulk Earth.

    eNd versus eSr - epsilon.gif

    A mixing line for this graph, e.g. MORB - Crust, can be calculated on the basis of the mixing equation:

[87Sr/86Sr]MIX = (X * [87Sr/86Sr]crust + (1-X) * [87Sr/86Sr]morb * Srmorbt/Srcrustt) / (X + (1 - X) * Srmorbt/Srcrustt)

    Problems: Given that rock sample A has 87Rb/86Sr = .04921 and 87Sr/86Sr(Total) = .70321, and the decay constant is 1.42x10(-11)/yr, if 87Sr/86Sr(Initial) of sample A = .7018, determine t, the age of the rock.

    Present day isotopic ratios for the Bulk Earth (mantle) are 87Sr/86Sr=.7045 and 87Rb/86Sr=.0827.

    If rock sample A was formed as a result of melting of the Bulk Earth mantle, determine the age of the melting event.

    Note: the 87Sr/86Sr of rock A and the Bulk Earth at the time of melting is not known. However, at the time of the melting event, the source mantle, the residual mantle, and the basalt will have the same 87Sr/86Sr ratio.

ans:  .70321 = Sr/Srt0 + .04921(e[1.42x10(-11)xt)] - 1)
ans:  .7045 = Sr/Srt0 + .0827 (e[1.42x10(-11)xt)] - 1)
ans:  .70321 = t0 + .04921(e[1.42x10(-11)xt)]) - .04921
ans:  .7045 = t0 + .0827 (e[1.42x10(-11)xt)]) - .0827
ans:  .00129 = .03349(e[1.42x10(-11)xt]) - .03349
ans:  .03478 = .03349(e[1.42x10(-11)xt])
ans: 1.038518961 = e[1.42x10(-11)xt
ans:             t = 2.66x10(9)

        Model the variation in e Sr87/86 and e Nd143/Nd144 between a continental source with Sr87/Sr86 = srsrg and Nd143/Nd144 = ndndg, and basalt with Sr87/Sr86 =srsrb and Nd143/Nd144 = ndndb; where Srbasalt/Srgranite= srbsrg and Ndbasalt/Ndgranite = ndbndg, in proportions ranging from 0 to 100 percent granite.

    Equations: srsrg = .706: srsrb = .702; srbsrg=.142857
    srsrmix(r) = (x*srsrg + (1-x)*srsrb*srbsrg)/(x+(1-x)*srbsrg)
    e srsr(r)  = (srsrmix(r)/.7045-1)*10000
    ndndg = .5124: ndndb = .5134;  ndbndg = .16666667
    ndndmix(r) = (x*ndndg + (1-x)*ndndb*ndbndg)/(x+(1-x)*ndbndg)
    e ndnd(r) = (ndndmix(r)/.51262-1)*10000
    Values used in the above equation are from Anderson, p. 202, Table 10-2.

        Excel solution is in: earthnt/public/300/isomix6.xls; c:\aacrse\300\rtf\isomix6.xls; c:\aacrse\330\ex\mix6.xls

        To learn more about mixing click here.


        Structural Provinces of North America.


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The Snowball Earth

       Carbon isotopic composition of Neoproterozoic glacial carbonates as a test of paleoceanographic models for snowball Earth phenomena.
    AU: Kennedy-Martin-J; Christie-Blick-Nicholas; Prave-Anthony-R
    SO: Geology (Boulder). 29; 12, Pages 1135-1138. 2001. .
    PB: Geological Society of America (GSA). Boulder, CO, United States. 2001.
    PY: 2001
    AB: Consistently positive carbon isotopic values were obtained from in situ peloids, ooids, and stromatolitic carbonate within Neoproterozoic glacial  successions in northern Namibia, central Australia, and the North American Cordillera. Because positive values continue upward into the immediately overlying postglacial cap carbonates, the negative isotopic excursions widely observed in those carbonate rocks require an explanation that involves a short-term perturbation of the global carbon cycle during deglaciation. The data do not support the ecological consequences of  complete coverage of the glacial ocean with sea ice, as predicted in the 1998 snowball Earth hypothesis of P.F. Hoffman et al. In the snowball
Earth hypothesis, the postglacial cap carbonates and associated -5% negative carbon isotopic excursions represent the physical record of CO2 transfer from the high-pCO (sub 2) snowball atmosphere ( approximately 0.12 bar) to the sedimentary reservoir via silicate weathering in the snowball aftermath. Stratigraphic timing constraints on cap carbonates imply weathering rates of approximately 1000 times preglacial levels to be consistent with the hypothesis. The absence of Sr isotopic variation between glacial and postglacial deposits and calculations of maximum weathering rates do not support a post-snowball weathering event as the origin for cap carbonates and associated isotopic excursions.

Discussion between Christie-Blick and Hoffman , March 1999 at

If highly depleted carbon isotopic values of cap carbonates are the result of the collapse of primary productivity, then maximum depletion of the ocean as a whole ought to date from the time at which the ocean was frozen. However, in Namibia (1, 5), isotopic depletion increases up section from the base of the cap carbonate (a trend that is typical of Marinoan cap carbonates) (5, 11). Hoffman et al. ascribe this trend to isotopic fractionation associated with the hydration of atmospherically derived CO2 in the surface ocean, with depletion returning to bulk oceanic values as the amount of CO2 in the atmosphere subsided from ~0.12 to 0.001 bar. This interpretation requires the ocean to have remained effectively lifeless for an unduly long span after snowball conditions had ceased--comparable to the duration of Marinoan deglaciation in Australia, including whatever time was needed for the drawdown of CO2 by continental weathering (104 to 106 years?) (12) and for deposition of the cap carbonates (<105 years) (13).

Christie-Blick et al. also question our interpretation of low carbon isotopic (13C) values in the cap carbonate above the glacial deposits, asserting that they require the ocean to be essentially lifeless for an extended time period after snowball conditions had ceased. The 13C value of marine carbonate reflects the relative amounts of carbonate carbon and organic carbon burial in sediments. In our hypothesis, the low 13C values reflect high rates of carbonate precipitation resulting from intense chemical weathering in the extreme greenhouse conditions following the melting of sea ice. If the rate of alkalinity delivery to seawater, and hence carbonate accumulation, was very high, recovery of biological productivity could be instantaneous after the deglaciation, and reach levels even greater than modern, but still not affect significantly the 13C values of the cap carbonates.

TI: Post-glacial carbonates of the Adrar region, Mauritania, and the snow-ball Earth hypothesis.
    AU: Shields-Graham-A
    BK: In: Geological Society of America, 1999 annual meeting.
    BA: Anonymous
    SO: Abstracts with Programs - Geological Society of America. 31; 7, Pages 487. 1999. .
    PB: Geological Society of America (GSA). Boulder, CO, United States. 1999.
    PY: 1999
    AB: In Mauritania, 7 m-10 m periglacial polygons cap Neoproterozoic-Cambrian tillites and represent the last traces of the cold, arid climate that led to continental glaciation across the whole of West Africa (Deynoux, 1980). Draping these polygons is found the thin, enigmatic dolostone that forms the subject of this presentation. The Jbeliat cap-dolostone is mechanically laminated with scoured bedding surfaces, and sheet, polygonal, and tepee-like dessication cracks. Barite is present as syndiagenetically contorted veins, cavern fills, crystal fans, and clusters of acicular crystals and is the subject of an ongoing geochemical study (Nd-Sr-C-O-S isotopes). Volcanically derived beds with marine calcite cements, glauconite and
phosphate occur above a significant hiatus. Below this hiatus, dolostones yield C-isotope values between -3.7 per mil and -2 per mil, while values are consistently positive above the hiatus. How does the Mauritanian cap bear on the snowball question? The snowball hypothesis (Hoffman et al., 1998) actually contains two quite different hypotheses: 1) equatorial glaciation, and 2) biopump failure (low C-isotope values). The West African craton is likely to have been at high southern latitudes and so has little relevance regarding the first hypothesis applied to this particular glaciation    (Marinoan?, Ediacarian?). Anomalously low C-isotope values from Mauritania are similar to published data from other post-glacial carbonates, but are also identical to seawater values from the early Cambrian that are not associated with faunal extinction, implying that other factors have been overlooked that might lower seawater C-isotope ratios. In future studies, it will be necessary to 1) correlate Neoproterozoic glaciations better (e.g., using Sr isotopes) so that we can be sure that we are comparing the same event, 2) constrain the length of the negative C-isotope excursion and the possible effect of ocean stratification on C isotopes, and 3) apply more sensitive geochemical proxies (e. g., Nd isotopes).

TI: Geochemical and isotopic implications of the snowball Earth hypothesis.
    AU: Schrag-Daniel-P; Hoffman-Paul-F; Bowring-Samuel-A
    BK: In: Geological Society of America, 1999 annual meeting.
    BA: Anonymous
    SO: Abstracts with Programs - Geological Society of America. 31; 7, Pages 372. 1999. .
    PB: Geological Society of America (GSA). Boulder, CO, United States. 1999.
    PY: 1999
    AB: The Snowball Earth hypothesis proposes that Neoproterozoic glacial deposits and associated "cap" carbonates represent a series of global glaciations followed by extreme greenhouse conditions. In the context of the hypothesis, a runaway ice-albedo feedback causes a global glaciation, with near-complete sea-ice cover, and a greatly reduced hydrologic cycle dominated by sublimation. Escape from this frozen state requires several to several 10's of millions of years for carbon dioxide, released by magmatic outgassing, to build up in the ocean/atmosphere system, providing adequate radiative forcing to overcome the high planetary albedo. Meltback would be extremely rapid (i.e., hundreds of years), transforming the
    earth from frozen to ultra-greenhouse conditions. The hypothesis predicts that the cap carbonates were rapidly deposited, with alkalinity supplied by intense carbonate and silicate weathering. An important question is whether carbonate dissolution during the glaciation was sufficient to maintain carbonate saturation. If so, then the rapid warming of the surface ocean would also drive massive carbonate deposition at a global scale, followed by continued deposition at lower latitudes due to weathering. The carbon isotopic compositions of the cap carbonates are consistent with this hypothesis. Values immediately on top of the glacial deposit are between 3 and 0 per mil, consistent with dissolved inorganic carbon in isotopic equilibrium with a CO (sub 2) -rich atmosphere. Values rapidly decrease to 5 per mil, consistent with Rayleigh distillation of the atmosphere as carbonate is deposited, and mass balance considerations. Elevated (super 87) Sr/ (super 86) Sr values above the basal carbonate unit are biased by
    in-situ Rb decay, but are consistent with very intense weathering of silicate rock flour after an initial sequence of carbonate deposition due to degassing of seawater during ocean warming and/or intense carbonate weathering prior to eustatic sea-level rise from melting continental glaciers. The reasons why the Earth was susceptible to such glaciations in the Neoproterozoic (and possibly the Paleoproterozoic) remains a mystery, but the assembly of large continents at low-latitudes may have been a contributing factor to achieving low atmospheric CO (sub 2) by reducing the negative feedback of ice-cover on silicate weathering of continents.

TI: Neoproterozoic low-latitude glaciation and the snowball Earth hypothesis.
    AU: Hoffman-Paul-F
    BK: In: Geological Society of America, 1999 annual meeting.
    BA: Anonymous
    SO: Abstracts with Programs - Geological Society of America. 31; 7, Pages 371-372. 1999. .
    PB: Geological Society of America (GSA). Boulder, CO, United States. 1999.
    PY: 1999
    AB: The occurrence of late Neoproterozoic glacial deposits on every continent led Harland (1964) to postulate a global ice age. Simple energy-balance climate models (e.g., Budyko, 1969) suggested that runaway ice-albedo feedback might occur if solar luminosity or greenhouse gas concentrations were substantially diminished. These findings were not taken seriously at first because there seemed to be no means of recovery from the high albedo of an ice-covered Earth and it was thought that all life would be extinguished. Caldeira and Kasting (1992) later estimated that recovery would be possible if atmospheric CO (sub 2) levels rose to approximately 0.12 bar (350x present), which could result from normal volcanic
outgassing over millions of years in the absence of sinks for carbon (i.e., no photosynthesis or silicate weathering). Reliable paleomagnetic evidence that ice lines reached sea level near the equator during the Marinoan ice age in Australia led Kirschvink (1992) to invoke an albedo-driven "snowball" Earth. He noted that global sea ice would limit air-sea gas exchange, leading to anoxic oceans rich in dissolved iron, explaining the co-occurrence of Neoproterozoic iron-formations and glacial deposits. My coworkers and I (1998) pointed out that petrographically distinctive "cap" carbonates and large negative d13C anomalies, both widely associated with Neoproterozoic glacial deposits, could be explained by a snowball Earth
and its ultra-greenhouse aftermath. Thus, the snowball Earth hypothesis is well grounded in theory (climate models), well supported by a variety of geological evidence (e.g., sea-level ice line near the equator, iron-formations with ice-rafted dropstones, "cap" carbonates with large negative d13Canomalies, large sea-level changes), and makes testable predictions concerning its longevity and its ultra-greenhouse aftermath. Moreover, as originally noted by Martin Rudwick (1964), the snowball hypothesis provides a new perspective on the longstanding problem of the origin of  metazoa. An evolutionary burst might be expected to result from the imposed series of population bottleneck-and-flush cycles, with severe genetic isolation during glaciations and unique transient selective environments at times of repopulation. The severity of these events may be judged from the long basal stem of eukarya in universal phylogenetic trees based on molecular sequencing.
    AN: 2001-023924

TI: The Paleoproterozoic snowball Earth; cyanobacterial blooms and the deposition of the Kalahari manganese field.
    AU: Kirschvink-Joseph-L; Gaidos-Eric-J; Beukes-Nic-J; Gutzmer-Jens
    BK: In: Geological Society of America, 1999 annual meeting.
    BA: Anonymous
    SO: Abstracts with Programs - Geological Society of America. 31; 7, Pages 372. 1999. .
    PB: Geological Society of America (GSA). Boulder, CO, United States. 1999.
    PY: 1999
    AB: Geological, geophysical, and geochemical data suggest that Earth experienced several intervals of intense, global glaciation ("snowball Earth" conditions) during Precambrian time, including at least one event in the Paleoproterozoic and perhaps four events during the Neoproterozoic. The abrupt, greenhouse-induced termination of these events would lead to the rapid deposition of both banded iron formations (BIFs) and cap carbonates. However, melting of the oceanic ice should also induce an immediate and massive bloom in the cyanobacteria, as deep-sea hydrothermal vent fluids are remarkably similar in composition to the nutrient media needed for cyanobacterial growth. This "green Earth" condition should produce an oxygen spike in the euphotic zone leading to the oxidative precipitation of ferric iron followed by manganese. We show that a particularly severe Paleoproterozoic snowball Earth at approximately 2.4 Ga would produce the geological pattern observed in the economically important Paleoproterozoic Kalahari Manganese Field (KMF) in Southern Africa. A newly-discovered drop-stone layer at the base of the Hotazel Formation (which contains the KMF) argues that the low-latitude glacial interval signaled by the Makganyene diamictite - Ongeluk volcanic sequence broke up just prior to KMF deposition. Due to the lower solar luminosity at this time, nearly 0.6 bar of CO (sub 2) would be needed in the atmosphere to break the snowball condition, which would require between 35 and 70 Myr to build up (at the present and twice the continental outgassing rates, respectively). If this scenario is correct, it represents a singular event in Earth history of a magnitude dwarfing later catastrophes such as the Cretaceous-Tertiary impact.