U-Pb geochronology using Pb-235U-(233U)-spiked … thermal ionization mass spectrometric data Mark D. Schmitz Department of Geosciences, Boise State University, 1910 University Drive,

Derivation of isotope ratios, errors, and error correlations forU-Pb geochronology using 205Pb-235U-(233U)-spiked isotopedilution thermal ionization mass spectrometric data

Mark D. SchmitzDepartment of Geosciences, Boise State University, 1910 University Drive, Boise, Idaho 83725, USA([email protected])

Blair SchoeneDepartment of Earth, Atmospheric and Planetary Sciences, Massachusetts Institute of Technology, Cambridge,Massachusetts 02139, USA

Now at Departement de Mineralogie, Universite de Geneve, Rue de Maraıchers 13, CH-1205 Geneva, Switzerland

[1] A comprehensive treatment of the derivation of U-Pb isotope ratios and their correspondinguncertainties from isotope dilution thermal ionization mass spectrometric measurements is presented.Standard parametric statistical methods of error propagation are utilized to convolve uncertaintiesassociated with instrumental mass fractionation, tracer subtraction, blank Pb and U subtraction, and initialcommon Pb correction. Derivations include errors and error correlations for total sample U/Pb and Pbisotope ratios (including radiogenic and initial common Pb) for two- and three-dimensional isochroncalculations, radiogenic U/Pb and Pb isotope ratios for concordia and radiogenic model age calculations,and the propagation of model age errors from radiogenic isotope ratios.

Components: 5621 words, 5 figures, 1 table.

Keywords: U-Pb; geochronology; error.

Index Terms: 1115 Geochronology: Radioisotope geochronology; 1194 Geochronology: Instruments and techniques; 1199

Geochronology: General or miscellaneous.

Received 26 September 2006; Revised 9 March 2007; Accepted 18 April 2007; Published 10 August 2007.

Schmitz, M. D., and B. Schoene (2007), Derivation of isotope ratios, errors, and error correlations for U-Pb geochronology

using 205Pb-235U-(233U)-spiked isotope dilution thermal ionization mass spectrometric data, Geochem. Geophys. Geosyst., 8,

Q08006, doi:10.1029/2006GC001492.

1. Introduction

[2] U-Pb accessory mineral geochronology utiliz-ing the isotope-dilution thermal ionization massspectrometry (ID-TIMS) method has the demon-strated potential to provide radiometric age con-straints for geological samples approaching, and

potentially exceeding the 0.1% level of precisionand accuracy. The analytical and instrumentalmethods of U-Pb ID-TIMS geochronology havereached their maturity over the past three decades,and have received intensive scrutiny regarding theappropriate methods of error propagation and as-signment to measured U-Pb (and Pb-Pb) ratios and

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their derivative ages [Cumming, 1969; Ludwig,1980; Mattinson, 1987]. The influential paper ofLudwig [1980] described a method for the propa-gation of errors associated with U-Pb ID-TIMSgeochronology. While the general nature of Lud-wig’s treatment has lent itself to widespread appli-cation over the past twenty-five years, advances inU-Pb analysis (e.g., reduction in analytical blankcontributions, the widespread availability and useof high-purity mixed 205Pb and double U ‘‘spikes’’or ‘‘tracers’’) have made some of the assumptionsof this treatment unnecessary or untenable, partic-ularly for very high-precision geochronologicalstudies. For example, although not treated byLudwig [1980], errors associated with the use ofa double U spike to correct instrumental isotopicfractionation should be accurately propagated intothe estimate of molar U quantities. Similarly, invery low-Pb samples (e.g., <10 pg radiogenic Pb)the uncertainties associated with the subtraction oftracer Pb isotopes may become a significant sourceof error in derived isotope ratios. Additionally, thesummary nature of Ludwig [1980] potentiallyobscures some of the underlying mathematics ofthe error propagation, particularly those in theisotope dilution calculations.

[3] We have revisited the problem of error propa-gation in U-Pb ID-TIMS mass spectrometry andgeochronology, with a specific emphasis on the useof a mixed 205Pb-235U-233U tracer, such as has beenrecently calibrated and distributed as part of theEARTHTIME Initiative for the sequencing of Earthhistory through the integration of high-precisiongeochronology and quantitative chronostratigraphy(http://www.earth-time.org). Our treatment isintended for both experts and novice users of U-Pbgeochronological data, thus we have attempted toclearly capture all of the algebraic manipulation andderivative calculus used in the derivations. Whilerecognizing that our statistical approach is notnovel, we hope that by providing a comprehensivederivation, we may elucidate the general method oferror propagation for students of geochronology,and inspire a similar degree of rigor in the propa-gation of errors for less mature U-Pb analyticaltechniques.

[4] The following sections detail the propagationof errors through the calculation of both radiogenicand sample (radiogenic + initial common) Pb andU/Pb isotope ratios, and radiogenic 206Pb/238U,207Pb/235U, and 207Pb/206Pb model ages, includingappropriate error correlations. The calculationsutilize the error propagation approximation result-

ing from the Taylor series expansion of the func-tion about its constituent variables (ignoringhigher-order terms) [Bevington and Robinson,1992], which in effect provides a linear extrapola-tion of the influences of arbitrarily small errors tothe real errors of interest:

x ¼ f u; v; . . .ð Þ ð1Þ

s2x ¼ s2

u �@x

@u

� �2

þs2v �

@x

@v

� �2

þ2 � s2uv �

@x

@u

� �� @x

@v

� �þ . . .

ð2Þ

where sx2 is defined as the variance of x, suv

2 isdefined as the covariance between u and v, and (@x/@u) is the first partial derivative of x with respect tou.

[5] In all of the following derivations, errors in thetracer Pb/U ratio (e.g., the moles of 205Pb and 235Uin the tracer) are considered systematic and thus areignored. Such errors are more appropriately eval-uated for the data set as a whole by propagation inquadrature on the weighted mean or similar groupstatistics [Schmitz et al., 2003]. On the other hand,errors in the tracer isotopic composition are prop-agated, as the effect of tracer subtraction on areduced isotopic ratio is dependent upon theamount of tracer added and the amount of eachisotope in the individual sample. The same is truefor the amount and isotopic composition of Pbblank contributions and the initial common Pbisotopic composition. The amount of both speciesof common Pb subtracted from an analysis issample dependent, and thus we consider that theerrors for each should be propagated on a sample-by-sample basis. However, we do advocate that todevelop a firm estimation of the sensitivity of adata set to the assumed initial common Pb compo-sition, the constituent data be reduced with a rangeof geologically reasonable initial Pb isotope ratios,or better a full Monte Carlo simulation of initial Pbisotope ratios, and the variance in resulting radio-genic model ages incorporated into the final ageinterpretation [Schmitz and Bowring, 2001;Schoene and Bowring, 2006].

2. Sample U-Pb Ratios, Errors, andError Correlations

[6] The following section details the propagation oferrors through the calculation of 205Pb-233U-235U-spiked sample U/Pb and Pb isotope ratios (where‘‘sample’’ molar quantities and ratios comprise both

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radiogenic and initial common Pb, but not tracer orblank Pb and U contributions), including errorsand error correlations associated with Pb and Uisotope fractionation corrections (including errorsin U isotope fractionation estimation utilizing adouble 233U - 235U tracer), and tracer and blankU and Pb subtractions. The resulting ratios, errorsand error correlations may be used in the calcula-tion of traditional 238U/204Pb-206Pb/204Pb,235U/204Pb-207Pb/204Pb, and 207Pb/204Pb-206Pb/204Pbisochrons, as well as two and three-dimensional iso-chrons utilizing 238U/206Pb-207Pb/206Pb-204Pb/206Pbratios [Ludwig, 1998].

[7] In the calculation of sample isotope quantities(e.g., 206Pb sample, 207Pb sample, 204Pb initial,238U sample) and U/Pb isotope ratios, most uncer-tainties in the constituent variables are assumed tobe uncorrelated. Exceptions are non-negligible er-ror correlations (covariances) between measuredisotope ratios in the expressions for 207Pb sampleand 204Pb initial. Similarly, in the calculation ofsample Pb isotope ratios, covariances betweennumerator and denominator must be calculatedand applied.

[8] The abbreviations listed in Table 1 are usedthroughout the derivations.

2.1. Derivation of Molar Isotope Quantitiesand Errors

2.1.1. Sample 206Pb

[9] First establishing the algebraic expression forsample 206Pb,

Pb206s ¼ R65m � Pb205t � 1þ FPbð Þ½ �� R65t � Pb205t½ � � Pb206b½ � ð3Þ

the error propagation equation may be written as(assuming all errors are uncorrelated):

s2Pb206s ¼

@Pb206s

@R65m

� �� sR65m

� �2þ @Pb206s

@R65t

� �� sR65t

� �2

þ @Pb206s

@FPb

� �� sFPb

� �2þ @Pb206s

@Pb206b

� �� sPb206b

� �2ð4Þ

[10] The partial derivatives are calculated asfollows,

@Pb206s

@R65m

� �¼ 1þ FPbð Þ � Pb205t ð5Þ

@Pb206s

@R65t

� �¼ �Pb205t ð6Þ

@Pb206s

@FPb

� �¼ R65m � Pb205t ð7Þ

@Pb206s

@Pb206b

� �¼ �1 ð8Þ

[11] These partial derivatives and variances canthen be substituted into equation (4) to derive theuncertainty in sample 206Pb.

2.1.2. Sample 207Pb

[12] First establishing the algebraic expression forsample 207Pb,

Pb207s ¼ R65m � R76m � 1þ 2 � FPbð Þ � Pb205t½ �� R65t � R76t � Pb205tð Þ � R76b � Pb206bð Þ ð9Þ

the error propagation equation may be written as(assuming all errors except R65m and R76m, andR65t and R76t are uncorrelated):

s2Pb207s ¼

@Pb207s

@R65m

� �� sR65m

� �2þ @Pb207s

@R76m

� �� sR76m

� �2

þ @Pb207s

@R65t

� �� sR65t

� �2þ @Pb207s

@R76t

� �� sR76t

� �2

þ @Pb207s

@FPb

� �� sFPb

� �2þ @Pb207s

@Pb206b

� �� sPb206b

� �2

þ @Pb207s

@R76b

� �� sR76b

� �2þ2 � s2

R65m�R76m

� @Pb207s

@R65m

� �� @Pb207s

@R76m

� �

þ 2 � s2R65t�R76t �

@Pb207s

@R65t

� �� @Pb207s

@R76t

� �ð10Þ

[13] The partial derivatives are calculated asfollows:

@Pb207s

@R65m

� �¼ R76m � 1þ 2 � FPbð Þ � Pb205t½ � ð11Þ

@Pb207s

@R76m

� �¼ R65m � 1þ 2 � FPbð Þ � Pb205t½ � ð12Þ

@Pb207s

@FPb

� �¼ 2 � R76m � R65m � Pb205t ð13Þ


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Table 1. Abbreviations

Abbreviation Description

GeneralFPb coefficient for linear Pb fractionation correction per unit mass differenceFU coefficient for linear U fractionation correction per unit mass differencel235 235U decay constantl238 238U decay constantt76 (207Pb/206Pb) radiogenic model aget75 (207Pb/235U) radiogenic model aget68 (206Pb/238U) radiogenic model age

Measured RatiosTracer quantities and ratiosa

R85m (238U/235U) measuredR35m (233U/235U) measuredR83m (238U/233U) measuredR65m (206Pb/205Pb) measuredR45m (204Pb/205Pb) measuredR75m (207Pb/205Pb) measuredR46m (204Pb/206Pb) measuredR76m (207Pb/206Pb) measured

Blank quantities and ratiosU238b moles (238U) blankPb206b moles (206Pb) blankR64b (206Pb/204Pb) blankR76b (207Pb/206Pb) blankR74b (207Pb/204Pb) blank

Initial Pb quantities and ratiosb

Pb206c moles (206Pb) initialPb204c moles (204Pb) initialR64c (206Pb/204Pb) initialR76c (207Pb/206Pb) initialR74c (207Pb/204Pb) initial

Sample quantities and ratiosU238s moles (238U) sampleU235s moles (235U) sampleR76s (207Pb/206Pb) sample (including radiogenic + initial Pb)R46s (204Pb/206Pb) sample (including radiogenic + initial Pb)R47s (204Pb/207Pb) sample (including radiogenic + initial Pb)R86s (238U/206Pb) sample (including radiogenic + initial Pb)R87s (238U/207Pb) sample (including radiogenic + initial Pb)R57s (235U/207Pb) sample (including radiogenic + initial Pb)R84s (238U/204Pb) sample (initial Pb)R54s (235U/204Pb) sample (initial Pb)Pb206s moles (206Pb) sample (including radiogenic + initial Pb)Pb207s moles (207Pb) sample (including radiogenic + initial Pb)

Radiogenic Pb quantities and ratiosPb206r moles (206Pb) radiogenicPb207r moles (207Pb) radiogenicR76r (207Pb/206Pb) radiogenicR68r (206Pb/238U) radiogenicR75r (207Pb/235U) radiogenicR86r (238U/206Pb) radiogenicR87r (238U/207Pb) radiogenic

a‘‘Tracer’’ is defined as an artificially produced or enriched isotope which is added to a sample for the purposes of concentration determination

through isotope dilution.b‘‘Initial’’ is defined as that Pb which was incorporated into the growing crystal at the time of formation.



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@Pb207s

@R65t

� �¼ �R76t � Pb205t ð14Þ

@Pb207s

@R76t

� �¼ �R65t � Pb205t ð15Þ

@Pb207s

@Pb206b

� �¼ �R76b ð16Þ

@Pb207s

@R76b

� �¼ �Pb206b ð17Þ

[14] The covariance terms can be determinedthrough the approximation for the error resultingfrom the product of two variables, x = uv, where forexample x = 207Pb/205Pb, u = 207Pb/206Pb, and v =206Pb/205Pb:

sx

x

� �2¼ su

u

� �2þ sv

v

� �2þ2 � s

2uv

u � v ð18Þ

[15] Solving for the covariance term:

s2uv ¼

u � v �$

sx

x

� �2� su

u

� �2� sv

v

� �2%

2ð19Þ

[16] Thus in this case:s2R76m�R65m

¼R76m � R65m �

$sR75m

R75m

� �2� sR76m

R76m

� �2� sR65m

R65m

� �2%

2

ð20Þ

s2R76t�R65t ¼

R76t � R65t �$

sR75t

R75t

� �2� sR76t

R76t

2� �

� sR65t

R65t

� �2%

2

ð21Þ

[17] These partial derivatives, variances and cova-riances can then be substituted into equation (10) toderive the uncertainty in sample 207Pb.

2.1.3. Sample (Initial) 204Pb

[18] First establishing the algebraic expression forsample (initial) 204Pb,

Pb204c ¼ R46m � R65m � 1� FPbð Þ � Pb205t½ �

� R46t � R65t � Pb205tð Þ � Pb206b

R64b

� �ð22Þ


s2Pb204c ¼

@Pb204c

@R65m

� �� sR65m

� �2þ @Pb204c

@R46m

� �� sR46m

� �2

þ @Pb204c

@R65t

� �� sR65t

� �2þ @Pb204c

@R46t

� �� sR46t

� �2

þ @Pb204c

@FPb

� �� sFPb

� �2þ @Pb204c

@Pb206b

� �� sPb206b

� �2

þ @Pb204c

@R64b

� �� sR64b

� �2þ2

� s2R65m�R46m � @Pb204c

@R65m

� �� @Pb204c

@R46m

� �

þ 2 � s2R65t�R46t �

@Pb204c

@R65t

� �� @Pb204c

@R46t

� �ð23Þ

[19] The partial derivatives are calculated as fol-lows,

@Pb204c

@R65m

� �¼ R46m � 1� FPbð Þ � Pb205t½ � ð24Þ

@Pb204c

@R46m

� �¼ R65m � 1� FPbð Þ � Pb205t½ � ð25Þ

@Pb204c

@FPb

� �¼ �R46m � R65m � Pb205t ð26Þ

@Pb204c

@R65t

� �¼ �R46t � Pb205t ð27Þ

@Pb204c

@R46t

� �¼ �R65t � Pb205t ð28Þ

@Pb204c

@Pb206b

� �¼ � 1

R64bð29Þ

@Pb204c

@R64b

� �¼ Pb206b

R64b2ð30Þ

and again the covariance terms can be calculated inthe manner of equations (18)–(19):

s2R65m�R46m ¼

R65m � R46m �$

sR45m

R45m

2� sR65m

R65m

2� sR46m

R46m

2%

2

ð31Þ



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s2R65t�R46t ¼

R65t � R46t �$

sR45t

R45t

2� sR65t

R65t

2� sR46t

R46t

2%

2ð32Þ

[20] These partial derivatives, variances, and cova-riances then can be substituted into equation (23) toderive the uncertainty in sample (initial) 204Pb.

2.1.4. Error in U Fractionation Factor (FU)for Double Spiked (233U-235U) Samples

[21] Because of the artificial nature of the 233Utracer isotope, a unique algebraic expression for thelinear uranium fractionation factor as the coeffi-cient per unit mass difference can be derived fromthe relationships:

R85f ¼ R85m � 3 � FU þ 1ð Þ ¼ U238sþ U238t

U235sþ U235tð33Þ

R35f ¼ R35m � 1� 2 � FUð Þ ¼ U233t

U235sþ U235tð34Þ

where R85f and R35f represent the mass fractiona-tion corrected ratios. Substituting the product(U235s � 137.88) for U238s (arising from thenatural 235U/238U ratio) and manipulatingequation (33):

U235s � R85m � 3 � FU þ 1ð Þ � 137:88½ �¼ U235t � R85t � R85m � 3 � FU þ 1ð Þ½ � ð35Þ

[22] Equation (34) can also be solved for U235s:

U235s ¼ U233t �$

1

R35m � 1� 2 � FUð Þ �1

R35t

%ð36Þ

[23] After substitution of equation (36) into (35),several cancellations result in the expression:

R35t � R85m � 3 � FU þ 1ð Þ � 137:88½ �¼ R85t � 137:88ð Þ � R35m � 1� 2 � FUð Þ½ � ð37Þ

[24] Expanding the products, gathering commonterms and solving for FU yields:

FU ¼ R35t � 137:88� R85mð Þ þ R35m � R85t � 137:88ð Þ2 � R35m � R85t � 137:88ð Þ þ 3 � R35t � R85m

� �ð38Þ

[25] The error propagation equation for FU may bewritten as (assuming all errors except R85m andR35m, and R85t and R35t are uncorrelated):

s2FU ¼ @FU

@R35t

� �� sR35t

� �2þ @FU

@R85m

� �� sR85m

� �2

þ @FU

@R35m

� �� sR35m

� �2þ @FU

@R85t

� �� sR85t

� �2

þ 2 � s2R85m�R35m � @FU

@R85m

� �� @FU

@R35m

� �

þ 2 � s2R85t�R35t �

@FU

@R85t

� �� @FU

@R35t

� �ð39Þ

[26] Through application of the chain and productrules, the partial derivatives are calculated as:

@FU

@R35t

� �¼ 137:88� R85m

2 � R35m � R85t � 137:88ð Þ þ 3 � R35t � R85m

� �

� 3 � R85m � R35t � R85m� 137:88ð Þ þ R35m � 137:88� R85tð Þ½ �2 � R35m � R85t � 137:88ð Þ þ 3 � R35t � R85m½ �2

!ð40Þ

@FU

@R85m

� �¼ �R35t

2 � R35m � R85t � 137:88ð Þ þ 3 � R35t � R85m

� �

� 3 � R35t � R35t � 137:88� R85mð Þ þ R35m � R85t � 137:88ð Þ½ �2 � R35m � R85t � 137:88ð Þ þ 3 � R35t � R85m½ �2

!ð41Þ

@FU

@R35m

� �¼ R85t � 137:88

2 � R35m � R85t � 137:88ð Þ þ 3 � R35t � R85m

� �

� 2 � R85t � 137:88ð Þ � R35t � 137:88� R85mð Þ þ R35m � R85t � 137:88ð Þ½ �2 � R35m � R85t � 137:88ð Þ þ 3 � R35t � R85m½ �2

!ð42Þ



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@FU

@R85t

� �¼ R35m

2 � R35m � R85t � 137:88ð Þ þ 3 � R35t � R85m

� �

� 2 � R35m � R35t � 137:88� R85mð Þ þ R35m � R85t � 137:88ð Þ½ �2 � R35m � R85t � 137:88ð Þ þ 3 � R35t � R85m½ �2

!

[27] The covariance terms are determined throughthe approximation for the error resulting from thequotient of two variables, x = u/v:

sx

x

� �2¼ su

u

� �2þ sv

v

� �2�2

s2uv

uvð44Þ

[28] Solving for the covariance term:

s2uv ¼

u � v � b su

u

2þ sv

v

2� sx

x

2c2

ð45Þ

[29] Thus in this case:

s2R85m�R35m ¼

R85m � R35m �$

sR85m

R85m

2þ sR35m

R35m

2� sR83m

R83m

2%

2

ð46Þ

s2R85t�R35t ¼

R85t � R35t �$

sR85t

R85t

2þ sR35t

R35t

2� sR83t

R83t

2%

2ð47Þ

[30] These partial derivatives, variances and cova-riances can then be substituted into equation (39) toderive the uncertainty in the uranium fractionationfactor.

2.1.5. Sample 238U

[31] First establishing the algebraic expression forsample 238U (making the trivial assumption thatblank 235U is negligible; the term (1/137.88) aris-ing from the natural 235U/238U ratio),

U238s ¼ R85m � 1þ 3 � FUð Þ � U235sþ U235tð Þ½ �� R85t � U235tð Þ � U238b

¼ R85m � 1þ 3 � FUð Þ � U238s � 1

137:88

� �þ U235t

� �� R85t � U235tð Þ � U238b

¼ U238s � 1

137:88

� �� R85m � 1þ 3 � FUð Þ

� �þ U235t � R85m � 1þ 3 � FUð Þ½ � � R85t � U235tð Þ � U238b

¼ U235t � R85m � 1þ 3 � FUð Þ½ � � R85t � U235tð Þ � U238b

1� 1137:88

� R85m � 1þ 3 � FUð Þ

�ð48Þ

the error propagation equation may be written as:

s2U238s ¼

@U238s

@R85m

� �� sR85m

� �2þ @U238s

@R85t

� �� sR85t

� �2

þ @U238s

@U238b

� �� sU238b

� �2þ @U238s

@FU

� �� sFU

� �2ð49Þ

[32] Note that to dramatically simplify our deriva-tion, we are considering error correlations betweenthe component terms of sample 238U (U238s) andthe uranium fractionation factor (FU) to be trivial,which is justified given the contrasting dominanterror sources in each quantity and the fact that wehave incorporated R85m-R35m and R85t-R35tcovariances into the error on FU. The partialderivatives are calculated as follows:

@U238s

@R85m

� �¼�

U235t � 1þ 3 � FUð Þ1� 1

137:88

� R85m � 1þ 3 � FUð Þ

� �

þ U235t � R85m � 1þ 3 � FUð Þ � R85t � U235tð Þ � U238b

1� 1137:88

� R85m � 1þ 3 � FUð Þ

�2"

� 1þ 3 � FU137:88

� ��ð50Þ

@U238s

@R85t

� �¼$

�U235t

1� 1137:88

� R85m � 1þ 3 � FUð Þ

%ð51Þ

@U238s

@U238b

� �¼$

�1

1� 1137:88

� R85m � 1þ 3 � FUð Þ

%ð52Þ

@U238s

@FU

� �¼$

3 � U235t � R85m1� 1

137:88

� R85m � 1þ 3 � FUð Þ

%

þ�U235t � R85m � 1þ 3 � FUð Þ � R85t � U235tð Þ � U238b

1� 1137:88

� R85m � 1þ 3 � FUð Þ

�2� 3 � R85m

137:88

� ��ð53Þ

[33] These partial derivatives and variances canthen be substituted into equation (49) to derivethe uncertainty in sample 238U.

ð43Þ



7 of 20

2.1.6. Sample 235U

[34] First establishing the simple algebraic expres-sion for sample 235U,

U235s ¼ 1

137:88

� �� U238s ð54Þ

the error propagation equation may be written as(assuming all errors are uncorrelated):

s2U235s ¼

@U235s

@U238s

� �� sU238s

� �2ð55Þ

[35] The partial derivative is simply:

@U235s

@U238s

� �¼ 1

137:88

� �ð56Þ

which can be substituted into equation (55) toderive the uncertainty in sample 238U. Note theresult that the relative error in sample 235U (U235s)is equivalent to the relative error in sample 238U(U238s).

2.2. Derivation of Sample IsotopeRatios and Errors

[36] The following derivations for sample isotoperatios and errors use the previously noted approx-imation for the error resulting from the quotient oftwo variables, x = u/v (equation (45)). All con-structed Pb isotope ratios are functions of one ormore common terms, and as such covarianceresulting from these common terms must be in-cluded in the error propagation. Analytical calcu-lation of covariance utilizes the followingrelationship [Bevington and Robinson, 1992]:

s2uv ¼ s2

c

@u

@c

� �@v

@c

� �ð57Þ

where u and v are functions of a common variable c.The total covariance is calculated as the sum of theshared individual covariances.

2.2.1. Sample 207Pb/206Pb

[37] We first recognize sample 207Pb/206Pb as thequotient of the previously derived terms, Pb206s

(equation (3)) and Pb207s (equation (9)), and thusdefine the sample 207Pb/206Pb variance as:

sR76s

R76s

� �2¼ sPb207s

Pb207s

� �2þ sPb206s

Pb206s

� �2� 2

Pb207s � Pb206s� s2

Pb207s�Pb206s

[38] The variance terms were previously calculatedfor Pb206s (equation (4)) and Pb207s (equation (10)).Noting that Pb206s and Pb207s are functions ofseveral common terms (R65m, R65t, FPb, Pb206b)the covariance between Pb207s and Pb206s iscalculated as:

s2Pb207s�Pb206s ¼ s2

R65m � @Pb207s

@R65m

� �� @Pb206s

@R65m

� �

þ s2R65t �

@Pb207s

@R65t

� �� @Pb206s

@R65t

� �

þ s2FPb �

@Pb207s

@FPb

� �� @Pb206s

@FPb

� �

þ s2Pb206b �

@Pb207s

@Pb206b

� �� @Pb206s

@Pb206b

� �ð59Þ

[39] Substituting the partial derivatives of Pb206s(equations (5)–(8)) and Pb207s (equations (11)–(17)) with respect to each common variable:

s2Pb207s�Pb206s ¼ R76m � 1þ 2FPbð Þ � Pb205tf g

� 1þ FPbð Þ � Pb205tf g � s2R65mR65t

2

þ 2 � R76m � R65m � Pb205tf g� R65m � Pb205tf g � s2

FPb

þ �R76bf g � �1f g � s2Pb206b ð60Þ

[40] This covariance can be substituted intoequation (58) to calculate the sample 207Pb/206Pbvariance.


[41] We first recognize sample 204Pb/206Pb as thequotient of the previously derived terms, Pb204c(equation (22)) and Pb206s (equation (3)), and thusdefine the sample 204Pb/206Pb variance as:

sR46s

R46s

� �2¼ sPb204c

Pb204c

� �2þ sPb206s

Pb206s

� �2� 2

Pb204c � Pb206s� s2

Pb204c�Pb206s ð61Þ

[42] The variance terms were previously calculatedfor Pb204c (equation (23)) and Pb206s (equation (4)).Noting that Pb204c and Pb206s are functions of

ð58Þ



8 of 20

several common terms (R65m, R65t, FPb,Pb206b), the covariance between Pb204c andPb206s is calculated as:

s2Pb204c�Pb206s ¼ s2

R65m � @Pb204c

@R65m

� �� @Pb206s

@R65m

� �

þ s2R65t �

@Pb204c

@R65t

� �� @Pb206s

@R65t

� �

þ s2FPb �

@Pb204c

@FPb

� �� @Pb206s

@FPb

� �

þ s2Pb206b �

@Pb204c

@Pb206b

� �� @Pb206s

@Pb206b

� �ð62Þ

[43] Substituting the partial derivatives of Pb206s(equations (5)–(8)) and Pb204c (equation (24)–(30)) with respect to each common variable:

s2Pb204c�Pb206s ¼ R46m � 1� FPbð Þ � Pb205tf g

� 1þ FPbð Þ � Pb205tf g � s2R65m

þ �R46t � Pb205tf g � �Pb205tf g � s2R65t

þ �R46m � R65m � Pb205tf g� R65m � Pb205tf g � s2

FPb

þ �1

R64b

� �� 1f g � s2

Pb206b ð63Þ

[44] This covariance can be substituted intoequation (61) to calculate the 204Pb/206Pb variance.Note that the relative error in sample 204Pb/206Pb isequivalent to the relative error in sample 206Pb/204Pb.


[45] We first recognize sample 204Pb/207Pb as thequotient of the previously derived terms, Pb204c(equation (22)) and Pb207s (equation (9)), and thusdefine the sample 204Pb/207Pb variance as:

sR47s

R47s

� �2¼ sPb204c

Pb204c

� �2þ sPb207s

Pb207s

� �2� 2

Pb204c � Pb207s� s2

Pb204c�Pb207s ð64Þ

[46] The variance terms were previously calculatedfor Pb204c (equation (23)) and Pb207s (equation(10)). Note that Pb204c and Pb207s are functionsof the common terms: R65m, R65t, F, Pb206b. Thecovariance between Pb204c and Pb207s is calcu-lated as:

s2Pb204c�Pb207s ¼ s2

R65m � @Pb204c

@R65m

� �� @Pb207s

@R65m

� �

þ s2R65t �

@Pb204c

@R65t

� �� @Pb207s

@R65t

� �

þ s2FPb �

@Pb204c

@FPb

� �� @Pb207s

@FPb

� �

þ s2Pb206b �

@Pb204c

@Pb206b

� �� @Pb207s

@Pb206b

� �ð65Þ

[47] Substituting the partial derivatives of Pb207s(equations (11)– (17)) and Pb204c (equations(24)–(30)) with respect to each common variable:

s2Pb204c�Pb207s ¼ R46m � 1� FPbð Þ � Pb205tf g

� R76m � 1þ 2 � FPbð Þ � Pb205tf g � s2R65m

þ �R46t � Pb205tf g � �R76t � Pb205tf g � s2R65t

þ �R46m � R65m � Pb205tf g� 2 � R76m � R65m � Pb205tf g � s2

FPb

þ � 1

R64b

� �� R76bf g � s2

Pb206b ð66Þ

[48] This covariance can be substituted into equa-tion (64) to calculate the 204Pb/207Pb variance.Note that the relative error in sample 204Pb/207Pbis equivalent to the relative error in sample207Pb/204Pb.

2.2.4. Sample 238U/206Pb, 238U/207Pb,238U/204Pb, 235U/207Pb, 235U/204Pb

[49] For the U/Pb ratios, errors in numeratorand denominator are considered essentially uncor-related, eliminating the covariance terms. Theresulting elementary expression for the propagatederror for the quotient of two uncorrelated parame-ters (x = u/v) is:

sRXYs

RXYs¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffisU23Xs

U23Xs

� �2þ sPb20Ys

Pb20Ys

� �2rð67Þ

where U23Xs, Pb20Ys, and RXYs are shorthand forthe necessary isotope quantities and ratios.

2.3. Derivation of Sample Isotope RatioError Correlations

[50] The correlation coefficient (r or rho) betweentwo isotope ratios is defined as the quotient of thecovariance and the product of the standard devia-tions for each ratio:

ruv ¼s2uv

susv

ð68Þ

[51] To solve for the correlation coefficient analyt-ically requires an estimate of the ratio covariance,for which we take advantage of the previouslynoted error equation for the quotient of two vari-



9 of 20

ables, x = u/v (equations (44)–(45)). Substitutingequation (45) into equation (68):

ruv ¼

u�v

$suuð Þ2þ sv

vð Þ2� sxxð Þ2%

2

su � sv

¼u � v � su

u

2þ sv

v

2� sx

x

2h i2 � su � sv

¼su

u

2þ sv

v

2� sx

x

22 � su

u

� sv

v

ð69Þ

[52] Using a simplified notation for the relativeerrors (or coefficients of variance):

%su ¼su

u; %sv ¼

sv

v; %sx ¼

sx

xð70Þ

(note that this definition is functionally acceptableas the factor of 100 percentage multiplier cancels inthe next expression) and substituting yields:

ruv ¼%s2

u þ%s2v �%s2

x

2 �%su �%sv

ð71Þ

2.3.1. Rho 238U/204Pb-206Pb/204Pb, Rho235U/204Pb-207Pb/204Pb, Rho238U/206Pb-207Pb/206Pb, and Rho238U/206Pb-204Pb/206Pb

[53] We can now define x, u, and v, and make theappropriate substitutions to define the equations foreach correlation coefficient. Defining U23Xs,Pb20Ys, Pb20Zs, and RXYs, RXZs, and RYZs asshorthand for the necessary isotope quantities andratios:

x ¼ RXYs ¼ U23Xs

Pb20Ysð72Þ

u ¼ RXZs ¼ U23Xs

Pb20Zsð73Þ

v ¼ RXZs ¼ Pb20Ys

Pb20Zsð74Þ

rRXZs�RYZs ¼%s2

RXZs þ%s2RYZs �%s2

RXYs

2 �%sRXZs �%sRYZs

ð75Þ

[54] In our specific cases:

rR84s�R64s ¼%s2

R84s þ%s2R64s �%s2

R86s

2 �%sR84s �%sR64s

ð76Þ

rR54s�R74s ¼%s2

R54s þ%s2R74s �%s2

R57s

2 �%sR54s �%sR74s

ð77Þ

rR86s�R76s ¼%s2

R86s þ%s2R76s �%s2

R87s

2 �%sR86s �%sR76s

ð78Þ

rR86s�R46s ¼%s2

R86s þ%s2R46s �%s2

R84s

2 �%sR86s �%sR46s

ð79Þ

2.3.2. Rho 207Pb/204Pb-206Pb/204Pb and207Pb/206Pb-204Pb/206Pb

[55] Similarly defining Pb20Xs, Pb20Ys, Pb20Zs,and RXYs, RXZs, and RYZs as shorthand for thenecessary isotope quantities and ratios:

x ¼ RXYs ¼ Pb20Xs

Pb20Ys¼ u

vð80Þ

u ¼ RXZs ¼ Pb20Xs

Pb20Zsð81Þ

v ¼ RYZs ¼ Pb20Ys

Pb20Zsð82Þ

rRXZs�RYZs ¼%s2

RXZs þ%s2RYZs �%s2

RXYs

2 �%sRXZs �%sRYZs

ð83Þ

[56] In our specific cases:

rR74s�R64s ¼%s2

R74s þ%s2R64s �%s2

R76s

2 �%sR74s �%sR64s

ð84Þ

rR76s�R46s ¼%s2

R76s þ%s2R46s �%s2

R74s

2 �%sR76s �%sR46s

ð85Þ

3. Radiogenic U-Pb Ratios, Errors,and Error Correlations

[57] The following section details the propagationof errors through the calculation of radiogenic206Pb/238U, 207Pb/235U, and 207Pb/206Pb ratios,including errors and error correlations associatedwith Pb and U isotope fractionation corrections(including errors in U isotope fractionation estima-tion utilizing a double 233U - 235U tracer), tracerand blank U and Pb subtractions, and initialcommon Pb corrections. The resulting ratios, errorsand error correlations may be used in the calcula-



10 of 20

tion and depiction of traditional Wetherill and Tera-Wasserburg type concordia diagrams, and radio-genic U-Pb and Pb-Pb model age calculations.

3.1. Derivation of Molar Isotope Quantitiesand Errors

3.1.1. Radiogenic 206Pb*

[58] First establishing the algebraic expression forradiogenic 206Pb,

Pb206r ¼ R65m � Pb205t � 1þ FPbð Þ½ �� R65t � Pb205t½ � � Pb206b½ � � R64c

��

R46m � R65m � Pb205t � R64c � 1� FPbð Þ½ �:

� R45t � Pb205tð Þ � Pb206b

R64b

� ��ð86Þ


s2Pb206r ¼

@Pb206r

@R65m

� �� sR65m

� �2þ @Pb206r

@R46m

� �� sR46m

� �2

þ @Pb206r

@R65t

� �� sR65t

� �2þ @Pb206r

@R45t

� �� sR45t

� �2

þ @Pb206r

@FPb

� �� sFPb

� �2þ @Pb206r

@Pb206b

� �� sPb206b

� �2

þ @Pb206r

@R64b

� �� sR64b

� �2þ @Pb206r

@R64c

� �� sR64c

� �2

þ 2 � s2R65m�R46m � @Pb206r

@R65m

� �� @Pb206r

@R46m

� �

þ 2 � s2R45t�R65t �

@Pb206r

@R45t

� �� @Pb206r

@R65t

� �ð87Þ

[59] The partial derivatives are then calculated asfollows:

@Pb206r

@R65m

� �¼ 1þ FPbð Þ � R64c � R46m � 1� FPbð Þ½ �

� Pb205tð88Þ

@Pb206r

@R46m

� �¼ �R64c � R65m � 1� FPbð Þ � Pb205t ð89Þ

@Pb206r

@R65t

� �¼ �Pb205t ð90Þ

@Pb206r

@R45t

� �¼ R64c � Pb205t ð91Þ

@Pb206r

@FPb

� �¼ 1þ R64c � R46mð Þ � R65m � Pb205t ð92Þ

@Pb206r

@Pb206b

� �¼ R64c

R64b� 1 ð93Þ

@Pb206r

@R64b

� �¼ �R64c � Pb206b

R64b2ð94Þ

@Pb206r

@R64c

� �¼ �R46m � R65m � Pb205t � 1� FPbð Þ½ �

þ R45t � Pb205tð Þ þ Pb206b

R64b

� �ð95Þ

[60] The covariance between R65m and R46m hasbeen derived in equation (31). The covariancebetween R65t and R45t is calculated by the methodof equations (44)–(45):

s2R45t�R65t ¼

R45t � R65t �$

sR45t

R45t

2þ sR65t

R65t

2� sR46t

R46t

2%

2ð96Þ

[61] These partial derivatives, variances, and cova-riances then can be substituted into equation (87) toderive the uncertainty in radiogenic 206Pb.

3.1.2. Radiogenic 207Pb*

[62] First establishing the algebraic expression forradiogenic 207Pb,

Pb207r ¼ R65m � R76m � Pb205t � 1þ 2 � FPbð Þ½ �� R65t � R76t � Pb205t½ � � R76b � Pb206b½ �� R64c � R76c � R46m � R65m � Pb205t½f

� 1� FPbð Þ� � R45t � Pb205t½ � � Pb206b

R64b

� ��

ð97Þ

the error propagation equation may be written as(assuming all errors except R65m, R46m andR76m, and R65t, R45t and R76t are uncorrelated):



11 of 20

s2Pb207r ¼

@Pb207r

@R65m

� �� sR65m

� �2þ @Pb207r

@R76m

� �� sR76m

� �2

þ @Pb207r

@R46m

� �� sR46m

� �2þ @Pb207r

@R65t

� �� sR65t

� �2

þ @Pb207r

@R76t

� �� sR76t

� �2þ @Pb207r

@R45t

� �� sR45t

� �2

þ @Pb207r

@FPb

� �� sFPb

� �2þ @Pb207r

@Pb206b

� �� sPb206b

� �2

þ @Pb207r

@R76b

� �� sR76b

� �2þ @Pb207r

@R64b

� �� sR64b

� �2

þ @Pb207r

@R76c

� �� sR76c

� �2þ @Pb207r

@R64c

� �� sR64c

� �2

þ 2 � s2R65m�R76m � @Pb207r

@R65m

� �� @Pb207r

@R76m

� �

þ 2 � s2R65m�R46m � @Pb207r

@R65m

� �� @Pb207r

@R46m

� �

þ 2 � s2R65t�R76t �

@Pb207r

@R65t

� �� @Pb207r

@R76t

� �

þ 2 � s2R45t�R65t �

@Pb207r

@R45t

� �� @Pb207r

@R65t

� �ð98Þ

[63] The partial derivatives are then calculated asfollows:

@Pb207r

@R65m

� �¼ ½R76m � 1þ 2 � FPbð Þ � R64c � R76c � R46m

� 1� FPbð Þ� � Pb205t¼ R76m � ð1þ 2 � FPb½ Þ � R74c � R46m� 1� FPbð Þ� � Pb205t ð99Þ

@Pb207r

@R76m

� �¼ R65m � 1þ 2 � FPbð Þ � Pb205t ð100Þ

@Pb207r

@R46m

� �¼ �R64c � R76c � R65m � 1� FPbð Þ � Pb205t

¼ �R74c � R65m � 1� FPbð Þ � Pb205tð101Þ

@Pb207r

@FPb

� �¼ 2 � R76mþ R64c � R76c � R46mð Þ � R65m

� Pb205t¼ 2 � R76mþ R74c � R46mð Þ � R65m � Pb205t

ð102Þ

@Pb207r

@R65t

� �¼ �R76t � Pb205t ð103Þ

@Pb207r

@R76t

� �¼ �R65t � Pb205t ð104Þ

@Pb207r

@R45t

� �¼ R64c � R76c � Pb205t ¼ R74c � Pb205t

ð105Þ

@Pb207r

@Pb206b

� �¼ �R76bþ R64c � R76c

R64b

¼ R64c � R76cR64b

� R64b � R76bR64b

¼ R74c� R74b

R64b

ð106Þ

@Pb207r

@R64b

� �¼ �R64c � R76c � Pb206b

R64b2¼ �R74c � Pb206b

R64b2

ð107Þ

@Pb207r

@R76b

� �¼ �Pb206b ð108Þ

@Pb207r

@R64c

� �¼ �R46m � R65m � Pb205t � R76c � 1� FPbð Þ½ �

þ R45t � Pb205t � R76cð Þ

þ Pb206b

R64b� R76c

� �ð109Þ

@Pb207r

@R76c

� �¼ �R46m � R65m � Pb205t � R64c � 1� FPbð Þ½ �

þ R45t � Pb205t � R64cð Þ

þ Pb206b

R64b� R64c

� �ð110Þ

[64] The covariance terms have been derived inequations (20), (21), (31) and (96). These partialderivatives, variances, and covariances then can besubstituted into equation (98) to derive the uncer-tainty in radiogenic 207Pb.

3.2. Derivation of Radiogenic IsotopeRatios and Errors

[65] As in section 2.2, the following derivations forradiogenic isotope ratios and errors use the approx-imation for the error resulting from the quotient oftwo variables (equation (45)). Radiogenic Pb/Uratios utilize the molar sample 238U and 235Uequations and errors derived in sections 2.1.5 and2.1.6. Covariance between numerator and denom-inator of the 207Pb*/206Pb* ratio is derived accord-ing to the method described in section 2.2.



12 of 20

3.2.1. Radiogenic 207Pb*/206Pb*

[66] We first recognize 207Pb*/206Pb* as a quotientof the previously derived terms, Pb206r (equation(86)) and Pb207r (equation (97)), and thus definethe 207Pb*/206Pb* variance as:

sR76r

R76r

� �2¼ sPb207r

Pb207r

� �2þ sPb206r

Pb206r

� �2� 2

Pb207r � Pb206r � s2Pb207r�Pb206r ð111Þ

[67] We have previously derived the variances forradiogenic 206Pb (equation (87)) and 207Pb(equation (98)). Note that Pb207r and Pb206r arefunctions of numerous common terms: R65m,R46m, R65t, R45t, FPb, R64b, Pb206b, R64c.The covariance between Pb207r and Pb206r isthus calculated as:

s2Pb207r�Pb206r

¼ s2

R65m � @Pb207r

@R65m

� �� @Pb206r

@R65m

� �

þ s2R46m � @Pb207r

@R46m

� �� @Pb206r

@R46m

� �

þ s2R65t �

@Pb207r

@R65t

� �� @Pb206r

@R65t

� �

þ s2R45t �

@Pb207r

@R45t

� �� @Pb206r

@R45t

� �

þ s2FPb �

@Pb207r

@FPb

� �� @Pb206r

@FPb

� �

þ s2Pb206b �

@Pb207r

@Pb206b

� �� @Pb206r

@Pb206b

� �

þ s2R64b �

@Pb207r

@R64b

� �� @Pb206r

@R64b

� �

þ s2R64c �

@Pb207r

@R64c

� �� @Pb206r

@R64c

� �ð112Þ

[68] The reader is referred to the derivations ofPb206r (section 3.1.1) and Pb207r (section 3.1.2)for the partial derivatives to substitute intoequation (112).

3.2.2. Radiogenic 206Pb*/238U, 207Pb*/235U,207Pb*/238U

[69] For the Pb*/U ratios (e.g., 206Pb*/238U,207Pb*/235U, 207Pb*/238U), errors in numeratorand denominator are considered essentially uncor-related, eliminating the covariance terms.

x ¼ u

v¼ Pb20Xr

U23Ys¼ RXYr ð113Þ

sRXYr

RXYr¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffisPb20Xr

Pb20Xr

� �2þ sU23Ys

U23Ys

� �2rð114Þ

where Pb20Xr, U23Yr, and RXYr are shorthand forthe necessary isotope quantities and ratios. Notethat the relative error in 206Pb*/238U is equivalentto the relative error in 238U/206Pb* for the purposesof representation in the Tera-Wasserburg concordiadiagram. Similarly, the relative error in 207Pb*/238Uis equivalent to the relative error in 238U/207Pb* forthe purposes of calculating error correlations forthe Tera-Wasserburg concordia diagram.

3.3. Derivation of Radiogenic Isotope RatioError Correlations

[70] The correlation coefficient (r or rho) betweentwo radiogenic isotope ratios is calculated accord-ing to the derivation of section 2.3. The followingerror correlations are applicable to the Wetherilland Tera-Wasserburg concordia diagrams.

3.3.1. Rho 207Pb*/235U-206Pb*/238U

x ¼ R76r ¼ Pb207r

Pb206rð115Þ

u ¼ R75r ¼ Pb207r

U235sð116Þ

v ¼ R68r ¼ Pb206r

U238sð117Þ

rR75r�R68r ¼%s2

R75r þ%s2R68r �%s2

R76r

2 �%sR75r �%sR68r

ð118Þ

3.3.2. Rho 238U/206Pb* - 207Pb*/206Pb*

x ¼ R87r ¼ U238s

Pb207rð119Þ

u ¼ R86r ¼ U238s

Pb206rð120Þ

v ¼ R76r ¼ Pb207r

Pb206rð121Þ

rR86r�R76r ¼%s2

R86r þ%s2R76r �%s2

R87r

2 �%sR86r �%sR76r

ð122Þ



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4. Radiogenic U-Pb and Pb-Pb Ages

4.1. Errors for 206Pb*/238U and 207Pb*/235UAges

[71] Recalling the solution to the decay equation,and corresponding age equation for the238U-206Pb* system:

R68r ¼ el238�t68 � 1 ð123Þ

t68 ¼ 1

l238

� �� ln R68r þ 1ð Þ ð124Þ

the error propagation equation may be written as(assuming l238 is a constant):

s2t68 ¼ s2

R68r �@t68

@R68r

� �2

ð125Þ

[72] Evaluating the partial derivative of t68 withrespect to R68r,

@t68

@R68r

� �¼ 1

l238

� �� 1

R68r þ 1

� �ð126Þ

and substituting the derivative results in the206Pb*/238U age error:

st68 ¼1

l238

� �� sR68r

R68r þ 1

� �ð127Þ

[73] By analogous derivation, the 207Pb*/235U ageerror is:

st75 ¼1

l235

� �� sR75r

R75r þ 1

� �ð128Þ

4.2. Error for 207Pb*/206Pb* Age

[74] Recalling the solution to the decay equationfor the 207Pb*-206Pb* system,

R76r ¼ 1

137:88

� �� el235�t76 � 1

el238�t76 � 1

� �

¼ 1

137:88

� �� el235�t76 � 1

� el238�t76 � 1 �1 ð129Þ

the error propagation equation may be written as(assuming l235 and l238 are constants),

s2R76r ¼

@R76r

@t76

� �� st76

� �2ð130Þ

which may be solved analytically for st76:

st76 ¼sR76r@R76r

@t76

� � ð131Þ

[75] In order to evaluate the partial derivative ofR76r with respect to t76, we apply the product rule:

@R76r

@t76

� �¼$

1

137:88

� �� el235�t76 � 1 %

� @

@t76el238�t76 � 1 �1h i

þ el238�t76 � 1 �1h i

� @

@t76

1

137:88

� �� el235�t76 � 1 � �

ð132Þ

[76] In order to evaluate the two constituent deriv-atives, we then apply the chain rule:

@

@t76

$1

137:88

� �� el235�t76 � 1 %

¼ 1

137:88

� �� l235 � el235�t76

ð133Þ

@

@t76el238�t76 � 1 �1h i

¼ �1ð Þ � el238�t76 � 1 �2

� l238 � el238�t76

ð134Þ

[77] Making the appropriate substitutions:

st76 ¼sR76r

1

137:88

� �� l235 � el235 � t76

� �el238�t76 � 1

2664

3775�

1

137:88

� �� l238 � el238 � t76

� �� el235�t76 � 1

el238�t76 � 1 2

2664

3775

ð135Þ



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[78] We note that this expression is equivalent toequation (3) of Ludwig [2000], without decayconstant errors.

5. Discussion

[79] An accompanying spreadsheet formulation ofthese derivations (auxiliary material1 Table S1)illustrates a number of analytical examples, includ-ing data for zircon and titanite from the OligoceneFish Canyon Tuff (FCT) [Schmitz and Bowring,2001], zircons from the Mesoproterozoic DuluthComplex Anorthosite Series (AS3) [Schmitz et al.,2003] and zircons from a Permo-Triassic tuff (SO3;J. Crowley, unpublished data, 2006). The followingdiscussion traces the contributions of various ana-lytical uncertainties to the propagated error in theradiogenic 206Pb*/238U ratio, in order to establishthe importance of each measurement or correctionto high-precision geochronology, as well as poten-tially guide future improvements in analytical pro-tocols and precisions. We leave it to the reader toapply a similar analysis to 207Pb*/235U and207Pb*/206Pb* ratio errors.

[80] While tracer subtraction has generally notbeen previously incorporated into error propaga-tion schemes, our analysis indicates that it can be anontrivial source of error for samples over-spiked(e.g., 206Pb/205Pb < 1) with a relatively impuretracer [Parrish and Krogh, 1987], if the relativelyhigh 206Pb/205Pb of the tracer is not preciselymeasured (e.g., to 0.01% or better). The magnitudeof the contribution of tracer 206Pb subtraction to the206Pb*/238U error is illustrated in Figure 1 for threescenarios using two tracers. The first two scenariosinvolve use of the aforementioned relatively im-pure 205Pb-rich spike (‘‘GSC’’), with two differentpropagated errors in tracer 206Pb/205Pb. Error con-tributions of >10% are evident for overspikedsamples assuming a tracer 206Pb/205Pb error of0.1%; fortunately the error contribution decreasesas the square of the assumed tracer 206Pb/205Pberror (Figure 1). The third scenario involves rela-tively imprecise knowledge of the tracer206Pb/205Pb ratio for a more pure 205Pb spike(‘‘ET’’), such as that used in the recently preparedEARTHTIME mixed Pb-U tracer solution. Figure 1clearly illustrates how the use of the pure spikemakes error contributions to the 206Pb*/238U ratioerror from spike subtraction trivial.

[81] Measurement errors in both Pb and U isotoperatios contribute a major component to the total206Pb*/238U error for all zircon analyses, as illus-trated in Figures 2, 3, and 4b. The errors onmeasured 206Pb/205Pb, 238U/235U, and 233U/235U(the last through the U fractionation correction,FU) are generally the largest subequal contributors,while error contribution from the measured204Pb/206Pb is proportionately smaller. Of the fourmeasured ratios, only the error in 204Pb/206Pb isuncorrelated with its percentage contribution to the206Pb*/238U error. Measurement errors for the otherthree ratios are strongly positively correlated withtheir contributions to the 206Pb*/238U error (Figure3). This fact emphasizes the importance of veryhigh precision isotope ratio measurements for pre-cise geochronology, either through static Faradaymeasurements (U and large Pb samples) or peakjumping on a linear, large dynamic range ion-counting system, ideally accommodating countrates to the megahertz (Mcps) range. In additionto the requisite precision, robust linearity over thisrange is clearly critical to the measurement accu-racy of the very large 206Pb/204Pb ratios (and verysmall 204Pb ion signals) of radiogenic samples.

[82] Fractionation correction error is a major con-tributor to the 206Pb*/238U error (Figures 2 and 4c),either manifested as a combination of measurementand tracer ratio errors for double-spiked U analy-ses, or as the reproducibility of an empiricalfractionation correction factor estimated from stan-dard Pb analyses. The latter is usually a moreimprecise quantity and thus larger contributor tothe total 206Pb*/238U error. This is particularly truewith the advent of U isotope analysis as the doubleoxide species using a silica gel emitter. The lowertemperatures and more stable ion currents of oxideanalyses result in a more reproducible mass frac-tionation; this fact is illustrated in Figure 2 by thecontrast in FU versus FPb error contributionsbetween the SO3 zircon analyses (UO2

+) and FCTand AS3 zircon analyses (U+). In the future,application of a double 202Pb-205Pb spike forinternal fractionation correction should also signif-icant decrease the FPb error contribution. We notethat the algorithms from this paper can be usedwith double Pb spikes as well, if the variance ofFPb is calculated internally using analogous meth-ods to those used for FU.

[83] Finally, we can assess the contributions oferrors associated with blank and initial commonPb subtraction (Figures 2 and 4a). In fact, due tothe nature of the partial derivatives of radiogenic

1Auxiliary material data sets are available at ftp://ftp.agu.org/apend/gc/2006gc001492. Other auxiliary material files are in theHTML.



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206Pb with respect to Pb blank amount and blankand initial Pb isotope composition, error contribu-tions from these three variables are relatively minorin the zircon analyses summarized in Figures 2 and4. Of the three quantities, Pb blank amount pre-dominates over a wide range of 206Pb/204Pb, yet forthe assumed blank amount in the range of 1–2picograms ±25% (1s) (the main group of analysesin Figure 4a) the contribution to 206Pb*/238U errorfrom blank amount uncertainty (being negativelycorrelated with 206Pb/204Pb) only exceeds 10% at206Pb/204Pb < 1000. The smaller group of analysesin Figure 4a with substantially lower contributionsto 206Pb*/238U error from blank amount uncertaintyare the SO3 zircons, which have significantlylower total common Pb and assumed blank amountuncertainty (average common Pb is 0.3 pg, andall common Pb is assumed to be blank), thusillustrating how modern low-blank analysis cansubstantially mitigate errors associated with com-mon Pb correction. Exploring this limit further, ifprocedural Pb blanks are reduced below 0.5 pg,then the analysis of only 5 pg of 206Pb* is requiredto essentially obviate errors associated with com-mon Pb correction. Figure 5 illustrates the combi-

nation of relevant factors including U content, ageand mass of zircon which yields the necessary 5 pgof 206Pb*.

[84] It is worth noting that blank and initial com-mon Pb composition uncertainties usually havecontributions to 206Pb*/238U error nearly twoorders of magnitude less than that contributed fromPb blank amount. However, an interesting cross-over in error contributions occurs at 206Pb/204Pbratios of approximately 100–200, whereby errorcontributions from uncertainty in initial commonPb composition begin to predominate over not onlythe other common Pb variables, but all sources oferror. This phenomenon is illustrated in Figures 2and 4a by Fish Canyon sphene analyses.

[85] Figures 4b and 4c illustrate how substantialdilution of the error contributions from most mea-surement errors (with the exception of R46m) andfractionation correction uncertainties only takesplace at relatively low 206Pb/204Pb ratios (<100).The obvious exception is the error contributionfrom R46m, which is inversely proportional to206Pb/204Pb (Figure 4b) as would be expected bythe increasingly important role this measured ratio

Figure 1. Contribution to radiogenic 206Pb*/238U error from tracer subtraction (e.g., tracer 206Pb) as a function ofmeasured 206Pb/205Pb for three scenarios and two tracer compositions. ‘‘GSC’’ represents a relatively high206Pb/205Pb tracer [Parrish and Krogh, 1987]; ‘‘ET’’ represents a lower 206Pb/205Pb tracer used in the EARTHTIMEmixed spike. Each data point represents an individual zircon or sphene analysis (see auxiliary material spreadsheetfile Table S1).



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Figure 2. Percentage contributions of all sources of analytical uncertainty to the radiogenic 206Pb*/238U error forfour large, high-quality data sets. Percentages as well as listed values and uncertainties for analytical parametersrepresent the median (measured or assumed) values for each data set; all analyses may be found in the auxiliarymaterial spreadsheet file Table S1. The median radiogenic 206Pb*/238U error of each data set is listed above eachchart. Assigned tracer composition uncertainties follow those of the ‘‘GSC_0.01%’’ scenario of Figure 1, resulting intheir negligible error contributions.

Figure 3. Contributions to radiogenic 206Pb*/238U error from isotope ratio measurement uncertainties as a functionof measured error for each ratio. Each data point represents an individual zircon or sphene analysis (see the auxiliarymaterial spreadsheet file Table S1).



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Figure 4. Contributions to radiogenic 206Pb*/238U error from uncertainties in (a) common Pb subtraction variables,(b) isotope ratio measurements, and (c) fractionation corrections, as a function of measured 206Pb/204Pb, a proxy forthe radiogenic/common Pb ratio. Each data point represents an individual zircon or sphene analysis (see the auxiliarymaterial spreadsheet file Table S1).



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plays for blank correction in low sample/blankanalyses [Mattinson, 1987].

[86] In summary, this analysis of error contributionto 206Pb*/238U ratio (and ultimately age) uncertain-ty not only quantitatively illustrates the necessarysample characteristics and mass spectrometricmethods required for U-Pb geochronology at thelevel of 0.1% age resolution, but also serves as atool to illuminate the way toward improvements inthat resolution through refinements in Pb fraction-ation correction, better mass spectrometry, andcleaner sample preparation.

Acknowledgments

[87] This contribution was catalyzed by discussions with

Felix Oberli on error propagation, and the ongoing efforts of

participants in the EARTHTIME project. We thank James

Crowley for the use of his unpublished data. Reviews and

editorial comments by Kenneth Ludwig, James Mattinson, and

Vincent Salters greatly improved the manuscript.

References

Bevington, P. R., and D. K. Robinson (1992), Data Reductionand Error Analysis for the Physical Sciences, 2nd ed., 328 pp.,McGraw-Hill, New York.

Cumming, G. L. (1969), A recalculation of the age of the solarsystem, Can. J. Earth Sci., 4, 719–735.

Ludwig, K. R. (1980), Calculation of uncertainties of U-Pbisotope data, Earth Planet. Sci. Lett., 46, 212–220.

Ludwig, K. R. (1998), On the treatment of concordant ura-nium-lead ages, Geochim. Cosmochim. Acta, 62, 665–676.

Ludwig, K. R. (2000), Decay constant errors in U-Pb concor-dia-intercept ages, Chem. Geol., 166, 315–318.

Mattinson, J. M. (1987), U-Pb ages of zircons: A basic exam-ination of error propagation, Chem. Geol., 66, 151–162.

Parrish, R. R., and T. E. Krogh (1987), Synthesis and purifica-tion of 205Pb for U-Pb geochronology, Chem. Geol., 66,103–110.

Figure 5. Graph of accumulated radiogenic 206Pb* per microgram of zircon versus age, illustrating the combinationof U content, zircon mass, and time necessary to produce 5 pg of radiogenic 206Pb*, which is a minimum amount toobviate errors associated with common Pb correction for a 0.5 pg procedural Pb blank. A single zircon grain 200 mmin long dimension with an aspect ratio of 2:1 weighs approximately 10 mg; thus all concentrations above 0.5 pg/mgzircon will produce >5 pg of radiogenic 206Pb* at ages older than the intersection of this concentration and theappropriate U concentration line (e.g., 7 Ma for a 500 ppm U zircon). A single zircon grain 100 mm in long dimensionwith an aspect ratio of 2:1 weighs approximately 1 mg; thus all concentrations above 5 pg/mg zircon will produce >5 pgof radiogenic 206Pb* at ages older than the intersection of this concentration and the appropriate U concentration line(e.g., 70 Ma for a 500 ppm U zircon).



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Schmitz, M. D., and S. A. Bowring (2001), U-Pb zircon andtitanite systematics of the Fish Canyon Tuff: An assessmentof high-precision U-Pb geochronology and its application toyoung volcanic rocks, Geochim. Cosmochim. Acta, 65,2571–2587.

Schmitz, M. D., et al. (2003), Evaluation of the Duluth Com-plex Anorthositic Series (AS3) zircons as a geochronological

standard: New high-precision IDTIMS analyses, Geochim.Cosmochim. Acta, 67, 3665–3672.

Schoene, B., and S. A. Bowring (2006), U-Pb systematics ofthe McClure Mountain syenite: Thermochronological con-straints on the age of the 40Ar/39Ar standard MMhb, Contrib.Mineral. Petrol., 151, 615–630.



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