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Algebraic and Graphic 

Methods for Evaluating 

Discordant Lead-Isotope 

Ages

GEOLOGICAL SURVEY PROFESSIONAL PAPER 414-E

Prepared on behalf of the U.S.

§ Atomic Energy Commission

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Algebraic and Graphic 

Methods for Evaluating 

Discordant Lead-Isotope 

Ages

By L. R. STIEFF, T. W. STERN, and R. N. EICHER

SHORTER CONTRIBUTIONS TO GENERAL GEOLOGY

GEOLOGICAL SURVEY PROFESSIONAL PAPER 414-E

Algebraic and graphic procedures for the calculation of a concordant age from discordant Pb206 /U 238, 

Pb207/U 235, and Pb207/Pb206 age data using one to three 

samples. Prepared on behalf of the U.S. Atomic 

Energy Commission

UNITED STATES GOVERNMENT PRINTING OFFICE, WASHINGTON : 1963
UNITED STATES DEPARTMENT OF THE INTERIOR 

STEWART L. UDALL, Secretary

GEOLOGICAL SURVEY 

Thomas B. Nolan, Director

For sale by the Superintendent of Documents, U.S. Government Printing Office

Washington 25, D.C.
CONTENTS

Abstract_ _ ___________________

Introduction ___________________

Previous work__________________

Graphic methods ___________

Algebraic methods __________

Method based on expansion of ex _.

Calculation of concordant ages___

One sample___-_-__________

Radiogenic Pb207/Pb206 ratio given_____________._____--____--_____--____-_______

Common Pb207/Pb206 ratio given________________________________________

Two samples._______________________________________________________________________

Amount of contaminating radiogenic lead and its Pb207/Pb206 ratio unknown. ____________

Amount of contaminating common lead and its Pb207/Pb206 ratio unknown.-.------------

Amount of contaminating radiogenic and common lead and the Pb207/Pb206 ratio unknownThree samples.-.-___________________________________________________________________

Amount and Pb207/Pb206 ratios of contaminating common and radiogenic lead unknown. __ 

General equations______________________________________________________________________

References cited_________________________________________________________________________

ILLUSTRATIONS

FIGURE 1. Ratios of the number of atoms of Pb207 to Pb204, N20T/N20t, plotted against the ratios of the number of atoms of 

Pb2°« to Pbm,Nm/Nm . _________________________________________________________________________

2. Ratios of the number of atoms of radiogenic Pb207 to U235, 2V_o7/N235, plotted against the ratios of the number of 

atoms of radiogenic Pb206 to U238, Nnt/Nna____----________________________________

3. Ratios of the total number of atoms of Pb207 to U235, W_07/JV.3Si plotted against the ratio of the total number of 

atoms of Pb206 to U238, Wm/Nm--------__________________________________________________

4. Normalized difference ratios for two uranium-bearing samples contaminated by the same common lead, and 

which were altered or contaminated by an older radiogenic lead._______________________________________

Page 

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2

2

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4

6

6

6

11

14

14

16

19

22

22

25

27

Page 

E3 

10 

19 

22

TABLES

TABLE 1. Letter symbols used in evaluating discordant lead-isotope ages.

2. Factors for use with equations of the second degree.__________

3-8. Computed concordant ages obtained 

3. From equation 12___________________________________.

4. From equation 17______________________________________.

5. From equation 22______________________________________.

6. From equation 26___ ________________________ _______.

7. From equation 35___-__________________________________.

8. From equation 39___-_________________________________

Page

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12

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24

m

SHORTER CONTRIBUTIONS TO GENERAL GEOLOGY

ALGEBRAIC AND GRAPHIC METHODS FOR EVALUATING DISCORDANT LEAD-ISOTOPE AGES

BY L. R. STIEFF, T. W. STERN, and R. N. EICHER

ABSTRACT

The calculated lead-isotope ages of many uranium-bearing 

minerals yield the following discordant age sequence: Pb206/ 

U238<Pb2°VU235 «Pb2<>VPb2 °6 . In the literature these and the 

reverse discordant lead-isotope age sequences are usually attributed to loss or gain of lead or uranium or an intermediate 

uranium daughter product. An alternative interpretation, 

generally not stressed, is that this failure of the lead-uranium 

and lead-lead ages to agree may also be a consequence of the 

deposition of an older generation of radiogenic Pb207 and Pb206 

at the time of formation of the uranium-bearing mineral. Also, 

the calculated lead-isotope ages of many uranium- and thoriumbearing minerals yield Pb208/Th232 ages which are less than the 

lead-uranium and lead-lead ages obtained on the same sample. 

The usefulness of the Pb208/Th232 age has been reduced because of the difficulties encountered in evaluating these age 

discrepancies.

Graphic solutions for the discordant lead-uranium age problem 

that are available in the literature have been amplified and extended to cover corrections for common lead without using 

either Pb204 or Pb208 as the common-lead index. Where isotope 

studies have been made on two or three samples of the same 

age and from the same deposit or formation, a new graphical 

procedure using the index isotope is presented that will yield 

concordant ages corrected for both lead loss and original radiogenic lead. These concordant ages may be obtained without 

knowledge of either the amounts or the isotopic compositions 

of the contaminating common lead, the Pb207/Pb208 ratio of 

original radiogenic lead, or the extent of past or recent alteration if original radiogenic lead is not present.

A new set of age equations has also been developed which 

permit algebraic solutions of the problems of original radiogenic lead and lead loss that are equivalent to the solutions 

obtained graphically. The derivation of these equations may 

also be considered as proof of the graphical constructions. A 

table of factors is provided in order that either exact lead-loss 

or original radiogenic lead solutions may be obtained from the 

age equations expanded only to the second degree. The general 

form of these new age equations is also given for programing 

on computing machines. The latter equations give both the 

concordant ages corrected for original radiogenic lead and for 

loss of lead. Both the graphic and algebraic solutions are 

applicable to the discordant thorium-uranium age problem.

The algebraic and graphic treatments of discordant leadisotope ages define both the type and minimum number of 

samples necessary for adequate mathematical analysis of the 

problem. This mathematical treatment also makes it clear that 

discordant lead-isotope age data alone cannot provide the basis 

for the choice of an age corrected for loss of lead or old radiogenic lead. The most reasonable age can be selected only after 

careful consideration of independent geochronologic data as well 

as field, stratigraphic, and paleontologic evidence, and the 

petrographic and paragenetic relations.

INTRODUCTION

Uranium-bearing minerals that give lead-uranium 

and lead-lead ages that are essentially in agreement, 

that is, concordant, generally are considered to have 

had a relatively simple geologic history and to have 

been unaltered since their deposition. The concordant 

ages obtained on such materials are, therefore, assumed 

to approach closely the actual age of the minerals. 

Many uranium-bearing samples, particularly uranium 

ores, give the following discordant age sequences: 

Pb206/U238<Pb207/U235<<Pb207/Pb206 or, less frequently, 

Pb207/Pb206 «Pb207/U235<Pb206/U238. In an effort to 

evaluate a discordant age sequence, therefore, the data 

are adjusted in one of several ways, either numerically 

or graphically, until the lead-uranium and lead-lead 

ages are in agreement. This is done in the belief that 

one of the recalculated concordant ages will more 

nearly approach the true age of the mineral. Thus, the 

criterion of concordance underlies the mathematical 

analysis of the observed age discrepancy and requires 

assumptions concerning the different processes which 

could have produced the age discordancies.

Unfortunately, the first discordant age sequence may 

be explained equally well by the continuous selective 

loss of one or more radioactive daughter products, by 

loss of radiogenic lead or additions of uranium at one 

time in the history of the minerals, or by initial contamination by relatively small amounts of an older 

generation of radiogenic lead. The reverse age sequence 

can be interpreted as evidence for loss of uranium, 

addition of lead, or initial contamination by relatively 

large amounts of an older radiogenic lead. Accidental 

concordance may occur either as a result of contamination by large amounts of an older generation of radiogenic lead, or as a result of processes of leaching and 

alteration in which compensating amounts of lead and 

uranium have been added or removed.

El
E2 SHORTER CONTRIBUTIONS TO GENERAL GEOLOGY

The evaluation of discordant lead-isotope age data 

may be separated into two operations. The first 

operation, with which this report is concerned, is 

mechanical in nature and involves the calculation of the 

different possible concordant ages corresponding to the 

various processes assumed to have produced the 

discordant ages. Present methods for making concordant numerical solutions of discordant lead-isotope 

age data are both tedious and difficult. Existing 

graphic solutions also have their limitations. This 

report includes a brief review of the literature on the 

graphical procedures used in the analysis of discordant 

age data and presents a new and more generalized 

graphical treatment of this problem. In addition, a new 

set of algebraic equations equivalent to these new 

graphic solutions is included. These equations permit 

relatively simple numerical calculation of the different 

and equally probable concordant ages. The letter 

symbols used throughout the report in presenting and 

analyzing the discordant age data are listed in table 1.

TABLE 1. Letter symbols used in evaluating discordant 

lead-isotope ages

Symbol

R.

Nd . 

NP . 

fiWn_

m.y.

Explanation

Number of atoms of Pb206, U238, and so on

Ratio of the number of atoms of radiogenic 

Pb207 to radiogenic Pb206

Ratio of the number of atoms of Pb207 to Pb206 

in common lead

Present-day ratio of the number of atoms of 

U235 to U238, 0.007262

Corrected concordant age of mineral. See 

equation 3

Decay constant

Age

Number of atoms of daughter products

Number of atoms of parent

Difference between two-term expansion of 

ex *i and the actual value of ex h for a particular value of t\, used for the U238 series

Difference between two-term expansion of 

ex h and the actual value of ex '2 for a particular value of t2 , used for the U235 series

Number proportional to the total number of 

radiogenic Pb206 atoms

Number proportional to the number of original 

radiogenic Pb206 atoms

Nth power

Number proportional to the total number of 

Pb206 atoms present

Ratio of *N, to N>,

Ratio of (N6 to N9

Million years

Ratio of the number of radiogenic Pb207 atoms, 

Nja, to the number of U235 atoms, Ns a, now 

present in sample A

It is evident, however, that in the first operation no 

amount of mathematical manipulation of the discordant 

age data will, in itself, provide the basis for the choice 

of the most probable age. The choice of one of the

equally possible recalculated concordant ages can be 

made only on the basis of additional evidence. The 

second operation, therefore, consists of testing the 

validity of these recalculated concordant ages in terms 

of the geologic history of the area; the geologic age 

relation of the enclosing rock; the petrographic, paragenetic, and mineralogic data on the uranium-bearing 

minerals being studied; the probable sources of the 

uranium and contaminating lead; the isotopic composition of lead in the associated nonradioactive 

minerals; and other independent age measurements 

which are considered to be reliable. In the event that 

available geologic evidence is equivocal, either the 

alternative concordant age solutions should be presented as equally possible or, following a clear statement 

of the investigator's own prejudice, a preference stated 

for one of the concordant ages.

General papers on the interpretation of discordant 

lead-isotope ages have been published by Kulp and 

others (1954), Ahrens (1955a, b), Wetherill (1956), 

Stieff and Stern (1956), Kulp and Eckelmann (1957), 

Aldrich and Wetherill (1958), and Stieff and Stern 

(1961). Explanations of discordant isotopic age data 

using the radon-loss hypothesis have been published by 

Wickman (1942), Robinson (1955), Louw and Strelow 

(1955), Giletti and Kulp (1955), and Greenhalgh and 

Jeffery (1959). Interpretations of discordant age 

sequences using the lead-loss hypothesis have been 

published by Collins and others (1954), Eckelmann and 

Kulp (1956), and Gerling (1958). Home and Davidson 

(1955) have suggested a hypothesis based on multiple 

periods of uranium deposition to explain the age 

anomalies found for a single specimen of uraninite 

concentrate from the Witwatersrand. Tilton (1960) 

has proposed continuous diffusion of lead as an explanation for discordant lead-isotope ages. Attempts 

to interpret the discrepancies between the lead-uranium 

and lead-lead ages on the basis of contamination by an 

older generation of radiogenic lead have been made by 

Stieff and others (1953), Tugarinov (1954), and more 

recently by Home (1957 a, b).

This work was part of a program conducted by the 

U.S. Geological Survey on behalf of the Division of 

Research, U.S. Atomic Energy Commission.

PREVIOUS WORK 

GRAPHIC METHODS

Perhaps the simplest graphical treatment of the 

regularity in lead-isotope data was suggested by 

Houtermans (1946, 1947). In an analysis of the 

natural variation in isotopic composition of common 

lead he plotted the atom or mole ratios of Pb207/Pb204i 

(NM/NMI) against similar ratios of Pb206/Pb204i 

(NzoB/Nzot). (See fig. 1A) In graphs of this type or
METHODS FOR EVALUATING DISCORDANT LEAD-ISOTOPE AGES E3

0

f/207

A/2Q4

0

A

^204

B

FIQUEE 1. Eatios of the number of atoms of Pba» to Pb^w, NmlNm, plotted against the ratios of the number of atoms of Pb^oe to Pb**, NmlNm. A, Two and three

component lead mixtures. B, Two component lead mixtures and their difference plots.

in graphs of Pb207/Pb208 versus Pb208/Pb208 (where 

thorium has not contributed radiogenic Pb208), the 

mixture of a single common lead, A, having uniform 

N2Q7/N2oi and Nm/N20i ratios, with varying amounts 

(points B, C, and D) of a single radiogenic lead of 

uniform Pb207/Pb208 ratio, R*, will lie on a straight line. 

These straight lines were called isochrones by Houtermans and the slope of the isochrones passing through 

these points is the Pb207/Pb208 ratio of the added 

radiogenic lead.

It is apparent from figure 1 that, if isotopic data 

are available for at least two radiogenically enriched 

samples, such as B and C, resulting from the mixture 

of only two components, R° and R*, the Pb207/Pb206 

ratio of the added radiogenic lead, R*, may be obtained 

from the slope of the line passing through B and C. 

Knowledge of the isotopic composition of the contaminating common lead, A, is not necessary. If the 

two components consist of the radiogenic lead produced 

by uranium-bearing minerals and its contaminating 

common lead, it follows that the slope of the line passing 

through the plotted points of the two samples will 

give the Pb207/Pb206 age of the samples. The quantitative amounts of lead and uranium do not appear

in these graphs; therefore, this age is independent 

of either recent loss or gain of lead or uranium. It is 

assumed, however, that isotopic fractionation of Pb206 

and Pb207 does not occur as a result of the chemical 

processes of alteration.

Figure 1A shows that the addition of varying

amounts of a third radiogenic component whose 

pb207/pb2o6 rati0j R) ^ give points E} F) an(j Q for

which no simple linear relation can be derived. The 

graphical presentation of the three component mixtures 

(Re , R*, and R) is typical of all discordant age sequences 

not produced by recent gain or loss of lead or uranium. 

The absence of any systematic relations between three 

or more samples, each of which contains a mixture of 

three different leads, suggests that additional data, 

perhaps the quantitative amounts of lead and uranium 

in the sample, may be required before a unique solution 

can be obtained.

In figure IB, the differences between the coordinates 

for samples (C minus B) and (D minus B) are plotted. 

These difference points for the two component mixtures, 

Re and R*, must also lie on a straight line passing 

through the origin. The slope of this line, R*, is 

equal to the slope of the line passing through the original
E4 SHORTER CONTRIBUTIONS TO GENERAL GEOLOGY

points B, C, and D. This analytical relation is 

mentioned here to simplify the presentation of some 

of the graphical procedures which follow.

Ahrens (1955b) noted the linear relations that are 

obtained in plots of the ratios of radiogenic N207/N235 

against radiogenic Nm/N2SS among related samples 

giving discordant age sequences. WetheriU (1956) 

developed Ahrens' suggestion into a rigorous graphical 

analysis of discordant lead-uranium ages resulting from 

multiple episodes of lead-uranium fractionation. The 

mathematical proofs of the properties of these ratio 

plots presented by Wetherill are not easy to follow, 

whereas the discussion of the concordia plots by 

Russell and Ahrens (1957) is quite brief. A relatively 

simple proof has been presented by Stieff and Stern 

(1961).

Finally, to Ahrens' plot of the mole ratios of N207/N235 

versus N2W/N238 Wetherill added a curve which he 

called concordia. (See fig. 2A.) Within this graph 

the coordinates of all points may be given by a Pb207/ 

U236 and a Pb206/U238 age equivalent to the two ratios, 

N2m/N235 and N206/N2S^. The curve, concordia, is the 

locus of all points having equal Pb207/U235 and Pb206/U238 

ages. Thus, a plot of the N207/N235 and N/206N23S ratios 

of any sample falling on this curve is, by definition, 

concordant, that is, the Pb206/U238 age=Pb207/U235 age= 

the Pb207/Pb206 age. Conversely, any point not lying 

on this curve must represent a discordant age sequence. 

It should be stressed that before it is possible to use a 

graph of this type, it is first necessary to make a correction for the contaminating common lead because the 

values A^o? and Nm represent only the radiogenic 

remainders. In those age calculations in which the 

common lead corrections are significant but have been 

incorrectly made, a discordant age sequence will result 

that is indistinguishable from those produced by the 

other causes already mentioned.

ALGEBRAIC METHODS

To the author's knowledge, only one published 

algebraic solution has been proposed for the originalradiogenic-lead problem. The lead-uranium isotope 

age equations originally derived by Keevil (1939),

and

* (yrs) =2.37X109 log

1 K p>v206

- iP

PK207

(1)

(2)

were modified by Tugarinov (1954) to include corrections for original radiogenic lead. The suggested 

modifications took the form of the following system of 

equations:

=2. 37X10. log

(3)

where Pb206 * and Pb207 * are the percent abundances of 

the original radiogenic lead, x is the percentage of the 

occluded original radiogenic lead in the uraniumbearing material, and tm is the corrected concordant 

age of the mineral.

Tugarinov's age equations, because of their log form, 

cannot be solved for x, the amount of original radiogenic lead present. In addition, it is necessary to know 

independently the Pb207/Pb206 ratio of the original 

radiogenic lead. Given Pb206 * and Pb207 *, repeated 

substitutions of different values for x must be made 

until a value of x is found which will make the Pb 206/U23& 

age equal to the Pb207/U23s age. The Pb207/Pb206 age 

corrected for the same amounts of original radiogenic 

lead will then be in agreement with the two leaduranium ages.

METHOD BASED ON EXPANSION OF ex

Although it is not difficult to obtain an approximately concordant age from discordant isotopic leaduranium data by the graphical methods discussed 

above, there are geologic problems for which the exact 

algebraic solutions may be preferred. Furthermore, as 

Aldrich and WetheriU (1958) have noted, "* * * the 

algebraic expressions involved are frequently so complex that it is difficult to visualize the physical processes 

[WetheriU (1956) and Wickman (1955)]." FinaUy, 

practical algebraic solutions provide an incentive to 

improve the measurement of the physical decay constants used hi age calculations, the analytical 

techniques, and sampling methods in order that we 

may derive aU of the useful geologic information that 

is available in a comprehensive lead-isotope age study. 

For these reasons, a set of algebraic expressions have 

been derived which are exact equivalents of the graphical 

procedures described by Stieff and Stern (1961). In 

these equations the role of the possible geologic processes can be visualized and it is possible to obtain as 

exact a concordant age solution as is required by the 

specific geologic problem.

An alternative approach to Tugarinov's algebraic 

solution of the problem of original radiogenic lead may 

be developed by using a series expansion of the exponential term in the general age equation where:

gX«__^y/^y _|_j (4) 

and where Nd and Np represent the number of atoms of
METHODS FOR EVALUATING DISCORDANT LEAD-ISOTOPE AGES E5

daughter product, D, and radioactive parent, P, 

respectively; X represents the decay constant in reciprocal years times 10~10 , and t, the age, in 1010 years 

equivalent to the atom ratio Na/Np. The value ex * 

may be approximated as closely as desired by the 

following series

2!

3! "

nl

Substituting this series for ex ', the age equation 4 may 

be written in the form

(W

2! 3! nl

,K .

(5)

For values of t that are less than 250 million years, 

equation 5 may be carried only to the second term 

without introducing serious errors. As t increases, 

however, it is necessary either to include a larger 

number of terms in the expansion of e and to work 

with equations of higher degree than two or to include 

an additional factor, /t or /2 , which may be obtained 

from table 2.

(6)

This factor/(li2) represents, for a number of values of 

ti, the difference between the two-term expansion of 

ex 'i for the U238 series, /i, and for the U235 series, /2, 

and the actual value of ex 'i for the particular value of ti. 

The decision to use equation 5 or 6 will be determined 

by the requirements of the specific geologic age problem, and by the section of the geologic time scale involved in the age calculations.

In the use of the second order equations, the selection of the initial age, ti, can be guided by the extent 

and type of discordant age sequences obtained in the 

trial age calculations. To calculate the age corrected 

for old radiogenic lead, the initial age ti and the corresponding values of/i and/2 from the table of factors 

are chosen so that ti is less than or equal to the trial 

Pb206/!!238 age. A selection of ti equal to or greater 

than the Pb^/Pb206 trial age will yield calculated ages 

corrected for loss or gain of lead or uranium. Should 

the nature of the geologic problem require a further 

refinement in the first calculated age, a second choice 

of ti based on the first calculated age, t, can be made. 

With experience in the initial choice of ti, it has been 

found that two age calculations will usually suffice 

for most geologic problems.

TABLE 2. Factors for use with equations of the second degree

tl

(millions of 

years)

0

20

40

60

80

100

120

140

160

180

200

210

220

230

240

250

260

270

280

290

300

310

320

330

340

350

360

370

380

390

400

410

420

430

440

450

460

470

480

490

500

510

520

530

540

550

560

570

580

590

600

610

620

630

640

650

660

670

680

690

700

710

720

730

740

750

760

770

780

790

800

810

820

830

840

850

860

870

880

890

U'38 series 

/i

O-OOOOo

.OOOOo

.OOOOo

.OOOOo

.OOOOo

.OOOOo

.OOOOo

.OOOOo

.OOOOo

.OOOOo

.OOOOo

.OOOOi

.OOOOi

.OOOOi

.OOOOi

.OOOOi

.OOOOi

.OOOOi

.OOOOi

.OOOOi

.00002

.00002

.00002

.00002

.00002

.OOOOs

.OOOOa

.OOOOs

.OOOOa

.00004

.OOOOi

.00004

.OOOOs

.OOOOs

.OOOOs

.OOOOe

.OOOOs

.OOOOe

.0000?

.0000?

.OOOOs

.00008

.00009

.00009

.OOOlo

.OOOlo

.OOOli

.OOOli

.00012

.0001s

.0001s

.00014

.0001s

.OOOle

.00016

.OOOl?

.0001s

.00019

.0002o

.0002o

.0002i

.0002s

.0002s

.00024

.0002s

.0002s

.0002?

.0002s

.OOOSo

.0003i

.00032

.0003s

.00034

.00036

.0003?

.00039

.0004o

.0004i

.0004s

.00044

U«3s series 

Jt

0. OOOOo

.OOOOo

.0000!

.OOOOs

.OOOOs

.00016

.0002?

.00043

.0006s

.0009s

.00128

.00149

.00172

.0019?

.00224

.00254

.0028:

.00322

.0036o

.0040i

.0044s

.00492

.00543

.0059?

.00654

. 00718

.0078i

. 0085o

. 00924

.OlOOi

. 0108s

. 01169

. 0126o

. 0135fl

. 01458

. 01562

. 0167s

.0178s

. 0191i

.02038

.0217i

. 0231o

.0245s

.02606

.02764

.0292s

. 03099

.03276

.0346i

.0365s

. 03852

. 0405s

.0427s

.0449s

.0472s

.0496s

. 0521o

.05464

. 0572g

.06001

.06282

. 0657s

.06874

.0718s

.0750s

.07832

.08172

.08522

.0888s

.09254

.09638

.10029

.10434

.1085o

. 1127s

. 11717

. 12169

. 12634

. 1311i

.1360o

tl 

(millions of 

years)

900

910

920

930

940

950

960

970

980

990

1000

1010

1020

1030

1040

1050

1060

1070

1080

1090

1100

1110

1120

1130

1140

1150

1160

1170

1180

1190

1200

1210

1220

1230

1240

1250

1260

1270

1280

1290

1300

1320

1340

1360

1380

1400

1420

1440

1460

1480

1500

1520

1540

1560

1580

1600

1620

1640

1660

1680

1700

1720

1740

1760

1780

1800

1820

1840

1860

1880

1900

1920

1940

1960

1980

2000

2020

2040

2060

2080

U*38 series 

/i

0.00046

.0004?

.00049

.0005o

.00052

.00054

.00056

.00057

.00059

.0006i

.0006s

.0006s

.0006?

.00069

.0007i

.0007s

. 0007s

.0007?

.OOOSo

.00082

.00084

.OOOSe

.00089

.0009i

.0009s

.00096

.00099

.OOlOj

.00104

.0010?

.OOllo

.00112

.0011s

.0011s

.0012i

.00124

.00127

.0013o

.00134

.0013?

,0014o

.0014?

.00154

.00160

.0016s

.0017s

.0018s

.0019i

.0020o

.0020s

.0021?

.00226

.0023s

.00244

.00254

.00264

.00274

.0028s

.00296

.0030?

.0031s

.00330

.0034]

. 00354

.00368

.00379

. 00392

.0040s

.00419

.0043s

.0044?

.00462

. 0047?

.00492

.0050s

.00524

.0054o

.0055?

.00574

.0059i

U*3« series 

ft

0. 14102

. 14619

. 15148

. 1569i

.1624s

.1681»

. 17404

. 18004

. 18619

.1924g

.1989s

.20554

.2123i

.21923

.22632

.23357

.24099

.24858

.25634

.26428

.2723»

.2806»

.2891s

.29784

.3067o

.31574

.32499

.3344s

.3440?

.3539s

. 3639s

. 37422

.3846g

. 3953g

.4062s

.4174o

. 42876

.4403s

. 45214

.4641s

.4764s

. 5017s

. 52799

. 5552s

.5835$

. 61296

.64344

. 67504

. 7078i

. 7417s

.7769o

. 81332

. 8510»

.8900o

.93034

.9720«

1. 0152

1. 059s

1. 105s

1. 1534

1.2026

1. 253s

1.305?

1. 359?

1. 415s

1.473o

1.532s

1. 593$

1. 6565

1. 721s

1.7884

1. 8574

1.9284

2.001s

2. 0768

2.1544

2.234i

2.316s

2.400?

2. 487.

649977 6S
E6 SHORTER CONTRIBUTIONS TO GENERAL GEOLOGY

TABLE 2. Factors for use with equations of the second degree Con.

(millions of 

years)

2100 

2120 

2140 

2160 

2180

2200 

2220 

2240 

2260 

2280

2300 

2320 

2340 

2360 

2380

2400 

2420 

2440 

2460 

2480

2500 

2520 

2540 

2560 

2580

2600 

2620 

2640 

2660 

2680

2700 

2720 

2740 

2760 

2780

2800 

2820 

2840 

2860 

2880

U»s series 

/i

0.00609 

.0062? 

.0064s 

.00664 

.0068s

.0070s 

.0072s 

.0074s 

.00764 

,0078s

.00806 

.0082s 

.0085i 

.0087s 

.00895

.0092o 

.00944 

.00968 

.00993

.01018

.01044 

.0107o 

.0109? 

.01124

.01179 

.01208 

.01236 

. 01266 

.01296

.01326 

.0135? 

.0138s 

.0142o 

.01452

.0148s 

. 0151s 

.01552 

.01586 

.0162i

TJ2*s series 

/i

2. 5769 

2.668s 

2.763s 

2.860s 

2.960s

3.0629 

3.1684 

3.276? 

3.388o 

3. 5024

3. 619s 

3. 740s 

3.8642 

3.99U 

4.1217

4. 2556 

4. 393i 

4.534! 

4.678s 

4.8273

4.979s 

5. 1358 

5.296o 

5. 460s 

5.628s

5. 8016 

5. 9786 

6.160? 

6.346s 

6.537j

6. 7326 

6. 932s 

7.138i 

7.3484 

7. 5639

7. 784e 

8. 010? 

8.242s 

8.479s 

8.7224

ti 

(millions of 

years)

2900 

2920 

2940 

2960 

2980

3000 

3020 

3040 

3060 

3080

3100 

3120 

3140 

3160 

3180

3200 

3220 

3240 

3260 

3280

3300 

3320 

3340 

3360 

3380

3400 

3420 

3440 

3460 

3480

3500 

3520 

3540 

3560 

3580

3600 

3620 

3640 

3660 

3680

U338 series

0. 01656 

.01692 

.01729 

.01766 

.0180s

.0184i 

.0188o 

.01919 

.01959 

.01999

.0204o 

.0208i 

.02124 

.02166 

.02209

.0225s 

.0229s 

.02343 

.0238s 

.02434

.0248i 

.02529 

. 0257? 

. 0262s 

. 0267s

.0272s 

.02776 

. 0282? 

.02879 

.0293i

.0298s 

.03039 

.0309s 

.0314g 

.0320s

.03262 

.03319 

. 0337? 

.03436 

.03496

U»» series 

/»

8.97h 

9.225s 

9.4866 

9. 7538 

10.02?

10.30? 

10. 593 

10.886 

11. 18» 

11. 49s

11.80s

12! 45s 

12.79s 

13. 139

13.492 

13. 852 

14. 22i 

14. 59s 

14. 984

15. 37s 

15. 782 

16. 19s 

16. 61? 

17.04s

17.49o 

17.94i 

18.40s 

18. 874 

19.356

19.849 

20.354 

20.869 

21.39s 

21.934

22. 484 

23. 046 

23. 62i 

24.20s 

24.80s

CALCULATION OF CONCORDANT AGES

ONE SAMPLE 

RADIOGENIC Pto*>i>/Pb206 RATIO GIVEN

If the Pb207/Pb206 ratio, R*, of the radiogenic lead 

lost at a time, t, or the contaminating original radiogenic 

lead is assumed or is inferred from independent evidence, 

such as the isotopic study of the lead in associated 

nonradioactive sulfides, it is possible to obtain an 

algebraic concordant age solution for a single sample 

without knowledge of either the amount of contaminating radiogenic Pb206 and Pb207 present in the 

sample or the extent of a single past period of alteration. 

However, this concordant age will be valid only if the 

following conditions are met.

1. The initial correction for common lead has been 

correctly made so that the remaining Pb 209 and 

Pb207 represent only the total radiogenic lead in 

the sample.

2. The sample has not selectively lost or gained radioactive daughter products since the time of mineral 

formation.

3. The sample has not been altered recently.

It is easiest to consider first the correction for 

original radiogenic lead. The solution for this problem

will also yield the concordant alteration age. Writing 

equation 5 for the Pb206/U238 and Pb207^235 ratios, 

respectively, one obtains the following:

n\

and

where

n\

(M) 2 , . ._N.-NS 

21

(X2 Q2

2!

(7)

(8)

\!=the decay constant for U238, 

X2=the decay constant for U235, 

N&= a number proportional to the total number

of radiogenic Pb206 atoms in sample A. 

NS*=B, number proportional to the number of

original radiogenic Pb206 atoms in sample

A. 

Nj=a number proportional to the total number

of radiogenic Pb207 atoms in sample A. 

Ni*=a, number proportional to the number of

original radiogenic Pb207 atoms in sample

A. 

N5=& number proportional to the number of

U235 atoms present now in sample A. 

N&=& number proportional to the number of

U238 atoms present now in sample A. 

£=age in years X1010

Equations 7 and 8 may be written in the form,

and

(9)

(10)

Dividing equation 10 by equation 9 one gets: 

'NT /(M)*

N5 wr\ri~ 2!

> r(r^r''' /(M)ft +_.i_(M) ~2T2 +Xl t_-y AV

and substituting the ratios one obtains:

 +

N?

R

\ n\ 2! CM)* 

nl +%?W)

=£' (11)

where

R*=N7 */N0 *t the ratio of the original radiogenic lead, 

and where

R=N5/N8=the present ratio of U235 to U238, a constant, 

= 1/137.7=0.007262!; 

and R^
METHODS FOR EVALUATING DISCORDANT LEAD-ISOTOPE AGES E7

Expanding equation (11) and collecting terms one 

obtains:

l V n\

(12)

If computing facilities are available, equation 12 

may be easily programed to solve directly for t. The 

degree of the equation used is determined in each case 

by the approximate age range of the sample being 

dated but generally will not exceed 10. For values of 

R*} R5, and Rs that are of geologic interest, equation 12 

has two positive real roots. As the degree of this equation increases, the smaller positive root rapidly approaches the concordant age, t corrected for original 

radiogenic lead as shown in figure 2B. The larger 

positive root approaches more slowly the value t as 

shown in figure 2A and is the concordant age obtained 

by correcting for loss or gain of lead or uranium at 

some time, ?i, in the past. In the latter case, R* is the 

Pb207/Pb206 ratio of the radiogenic lead produced in 

the time interval t to ti and lost at the time, t\.

If computing facilities are not available, equation 12 

may be taken only to the second power of t. As in 

equation 6, when t is greater than 250 million years 

factors /x and /2 should be added for the U238 and U235 

series, respectively. These factors obtained from 

table 2 for different values of ti, compensate for the 

failure of this second degree equation to approximate 

closely enough the actual value of ex ' for a specific 

value of t.

Equation 12 then takes the form

-Rf2)=Q. (13)

Solving for t,

t=-

where 2a= (R*\t- R\$)

2a

c= (RR5 -R*R& -RJ2+R%')

and where

/!=the factor for the U238 series depending on ti

/2=the factor for the U235 series depending on £,.

For values of R*, R5, and R8 that are of geologic

interest, equation 13 also has two positive real roots.

In using equation 13, the smaller root can be made to 

approach the concordant age corrected for original 

radiogenic lead by selecting values of fa and /2 from 

table 2 for ti that are less than the trial Pb206/!!238 age. 

With even approximately correct values for ji and /2, 

the smaller root will fall close to the actual age of the 

sample corrected for original radiogenic lead. The 

larger root, however, depending on the extent of the 

lead-uranium age discordancy, may depart substantially from the concordant 'lead-loss age."

If it desired to calculate exactly the alteration or 

"lead-loss age" using equation 13, values of/x and/2 

must be chosen from the table for ti equal to or greater 

than the Pb207/Pb206 age. Because the terms /t and /2 

become increasingly important as t increases, it may be 

desirable to make a rough graphical lead-loss solution 

to guide the initial choice of ti.

The departure of the first calculated age, t, from the 

exact lead-loss solution may be tested by comparing 

the calculated value of R* with the given value of 

R* (using equation 11). The first calculated value of 

t from equation 13 along with the appropriate values 

of /i and /2 for this calculated value of t from table 2 

may be substituted in equation 11. Wickman, 1939, 

Kulp and others (1954), Greenhalgh and Jeffery (1959), 

and Stieff and others (1959) have prepared tables for 

the calculation of lead isotope ages. If the tables by 

StieflF and others (1959), (hereinafter referred to as the 

age tables) are used, the terms [fi+(Xi£) 2/2+Xi£] and 

f/2+(X2£) 2/2+X2£] may be replaced by values for the 

N2o6/N238 and N207/N235 ratios for the appropriate value 

of t. If for the particular geologic problem the agreement between the calculated R* and the given R* is 

not satisfactory, a new ti is chosen and the lead-loss 

age is recalculated. This age calculation may be 

repeated until the desired degree of agreement between 

the calculated R* and the given R* is obtained.

EXAMPLE OF CAICUIAHOBT

An example of the use of equations 12 and 13 for 

the calculation of a concordant age corrected for 

original radiogenic lead is given below. The following 

isotopic data are available for a hypothetical unaltered 

sample of uraninite, the true age of which is 200 

million years and which contains (chemical scale) 1.144 

percent Pb and 27.923 percent U:

Pb2<"

83.936 7. 8. 203

Isotopic composition of lead 

from uraninite (in atom 

percent)____--_-_-_---_ 0.2125

Assumed isotopic composition of associated common lead (in atom percent).. _.__________ 1. 356 25. 044 21. 244 52. 35,

Ratio of original radiogenic lead present=Ni*/N8* R*= 

as found in associated nonradioactive mineral.
E8 SHORTER CONTRIBUTIONS TO GENERAL GEOLOGY

Using Pb204 as the "index" of the amount of common 

lead present and correcting for the proportionate 

amounts of nonradiogenic Pb206, Pb207, and Pb208, the 

remaining atom percent of radiogenic Pb208 and Pb207 

may be obtained as follows:

Isotopic analysis of uraninitelead___________ 0. 2125

Isotopic analysis of common lead___________ 0. 2125

Pb206

83. 936

3. 923

Pb208

7. 64e 8. 203 

3. 328 8. 203

Radiogenic lead produced

by uranium___________ 0.000 80. 01 3 4. 31g 0.000

In order to use equations 12 or 13, it is necessary to 

express the chemical lead and uranium data as quantities directly proportional to the number of atoms of the 

particular radioactive parent or radiogenic daughter 

isotope in the sample at the present tune. For exact 

solutions, the quantitative lead analyses, usually reported in the chemical scale, must be corrected for the 

actual difference in the physical atomic weight of the 

radiogenic lead in the sample. In this example it is 

206.27, compared to the assumed average physical 

atomic weight, 207.28, used in the chemical analysis. 

The correction factor for this particular case is 207.28/ 

206.27. Failure to correct the quantitative chemicallead data for the actual atomic weight of the radiogenic 

lead in the sample will introduce additional errors of 

0.2 to 0.5 percent in the final calculated lead-uranium 

ages.

This corrected weight must now be multiplied by the 

atom-percent abundance of the radiogenic Pb206 or Pb207 

in the sample after correction for common lead and divided by the atomic weight of the radiogenically enriched lead in the sample to get a quantity exactly 

proportional to the number of atoms present. Although 

it is necessary to include both Avogadro's number and 

the conversion factor from the chemical to the physical 

scales to obtain numbers of atoms, both of these terms 

cancel out in the ratio of the number of atoms of lead to

uranum.

1.144X207.28X80.01 3 

6 206.27X206.27 "-

1.144 X207.28X4.318 

206.27X206.27 =0 ft9 -0240«

N7 may be more easily obtained by multiplying N6 by the 

ratio of the radiogenic Pb207 to Pb206 remaining after correction for common lead, that is, 0.44594 X sTTrrr =0.02406 .

oO.Ulg

Similarly, N5 and N$ may be obtained by multiplying 

the percent uranium (chemical scale) by the atom-percent abundance of the U235 or IP38, and dividing by the 

physical atomic weight of U, 238.103.

27.923 X99.279 

238.103

27.923 X0.7209 

238.103

=ll.b42

=0.084546

N5 may also be obtained by dividing N& by 137.7, 

the atom ratio of U238 to U235 ; that is, 11.643 -5-137.7 

=0.084546 .

Expressing these analytical data as ratios of the number of atoms of radiogenic lead and uranium and using 

the age tables, we obtain the following ratios and discordant trial ages:

Ratio Trial age (millions of years)

0. 038307 244 (Pb208/U238)

. 28458 258 (Pb2<>VU235)

. 0539s 380 (Pb20VPb208)

If, in the course of the evaluation of this discordant age 

sequence, it is now desired to test the assumption, 

among others, that an additional correction for original 

radiogenic lead should have been made; equation 12 can 

be used to obtain a single corrected concordant age, t. 

As has been mentioned, equation 12 has been programed for the Geological Survey's digital computer. 

The values for R5, R%, R* and R given for this example, 

when substituted in equation 12, yield two positive real 

roots of t (in millions of years) as the degree of the 

equation increases from 2 to 10 (table 3). In this example, even the second degree equation yields a value 

for the smaller positive real root that closely approximates the hypothetical age of 200 million years. For 

such young ages, an equation of the third degree would 

give an age corrected for original radiogenic lead with 

less uncertainty than the uncertainty introduced by the 

experimental errors. The larger positive root converges 

more slowly on the "lead loss" age. An equation of the 

fifth degree, however, gives a geologically useful answer.

TABLE 3. Computed concordant ages obtained from equation 12

Degree of equation

2___._ -_- 

3_____-_-__-__-___---_-

4____ _ __.._ .

5 _ _

6___._ -_-- -___-

7-.-. ------------------

8___. _-_ _ _.

9.. _____ _ __ _ ________

10__._. ________________

Age (millions of years)

Corrected for original 

radiogenic lead

197.34 

200 2 4

200 so 

200.40 

200.40 

200.40 

200.40 

200 40 

200.40

Corrected for loss or 

gain of lead or uranium

1344.5 

908 39 

860.77

853.02 

853 oo 

852.89 

852.88 

852.88 

852 ss
METHODS FOB EVALUATING DISCORDANT LEAD-ISOTOPE AGES E9

The decision to include the terms R*fi and Rf2 in 

equation 13 is determined both by the nature of the 

problem and by the approximate range in age of the 

sample being dated. An examination of table 2 for 

values of t\ in the range of 250 million years suggests 

that both /i and /2 can be initially neglected. This 

decision is further supported by the fact that for this 

hypothetical problem, a calculated age within five 

percent of the actual age, and an agreement between 

the Pb206/U238 and Pb207/U235 ages of within one percent 

would be more than adequate. The ratio of the original 

radiogenic lead, R*, used in equation 13 may be 

assumed, or, as in this example, it may be obtained from 

the isotopic study of the lead extracted from associated 

nonradioactive minerals. Substituting in 13 the values 

5=1/137.7=0.007262!, #5=0.28458 , #s =0.038307, and 

.R*=--0.07185, where

2a= (R^l-R^l) X 10-20 y-1

= (0.07185 X2.362 1 -0.0072621 X94.510)X10-2(V~1

= (0.07185 X1.5369 -0.007262 1 X9.7216)X10-1 °2 

=0.039826 X10-10 2T1

= (0.007262 1 X0.28458 -0.07185 X0.038307) 

= -0.0006857

t becomes

 b±Jb2 -4ac t--

2a

~L -0.5166 _\ -0.039826 ±V.001586i-.0007084rL, nlo 2 JX1° y

r-0.039826 ±V.0008776rL u 

 n ^1 r f ^

H;T!!&ir*>"*

Z=0.01974 X1010=197 m.y. (age corrected for old

radiogenic lead)

£=0.13443 X1010= 1344.3 m.y. (age corrected for loss

or gain of lead or 

uranium).

These two concordant ages are in close agreement with 

the solutions for equation 12 expanded to the second 

degree (table 3).

For most geologic problems falling in this age 

range, the value of £=197 m.y. would be acceptable; 

the age would be rounded to the nearest 5 million years, 

that is, 195 million years. If, however, a closer approxior

mation of the true age is required, a test of the convergence of the lead-uranium and lead-lead ages 

may be made by first calculating N*6 using equation 7 

and Z=0.0197X1010 y. If the age tables are used, the

sum of the terms

(M)"

n\ +' 2! ' be replaced with the appropriate value of N2G&/N208 for t= 

0.0197X1010 years. If the age tables are not used, the 

term /i for the closest value of ti from table 2 must be 

added to the two-term expansion of eXl ', that is,

Given the ratio N*/N*6 =R*=.07185} when N*Q = 

0.08800 , N*7 is found to be 0.006323 . The Pb2o6/U238, 

Pb207/U235 and Pb207/Pb206 ages corrected for original 

radiogenic Pb206 and Pb207 can be obtained now by 

subtracting N* and N* from the total number of atoms 

of N& and N7} respectively. The recalculated ages are:

197 m y. 196 m.y. 184 m.y.

As the Pb206/U238 and Pb2o7/U235 ages are nearly in agreement and because the Pb20VPb206 age is less than the 

two lead-uranium ages, a slightly larger value is chosen 

for ti, that is, £i=198 m.y. In table 2, it is noted that 

for ^=198 m.y.,/! is negligible and/2 =0.00125.

The value of the coefficient c in equation 13 now becomes 0.0006932 , where RJ2=0.0000091, while the 

coefficients a and b in equation 13 remain unchanged. 

Using these coefficients in equation 13 the new age is 

found to be, £=200.o m.y. For this value of t, it can 

be shown that the lead-uranium and lead-lead ages 

converge exactly:

200 m.y. 200 m.y. 200 m.y.

It is thus possible, as part of the evaluation of a 

discordant age sequence, to calculate the single concordant age that would result from an additional 

correction for original radiogenic lead of known ratio. 

It is not necessary that the amounts of original radiogenic Pb206 and Pb207 be known. In this case, however, 

it is necessary that the proper common-lead correction 

be made, and that the sample be unaltered. This 

corrected age may now be compared with similar 

calculations on other samples from the same area or 

deposit. Finally, the ages obtained assuming correction for original radiogenic lead as well as losses or 

gains of lead and uranium must be combined with 

both field and other laboratory data to establish the 

most consistent interpretation of all the evidence, 

isotopic as well as geologic.
E10 SHORTER CONTRIBUTIONS TO GENERAL GEOLOGY

FIGURE 2. Ratios of the number of atoms of radiogenic Pb207 to U23S , NmlNta, plotted against the ratios of the number of atoms of radiogenic Pb2M to U238, Niw/Nm. Left, 

Uranium-bearing samples which lost or gained lead or uranium at a time, h, in the past. Right, Uranium-bearing samples contaminated with an original radiogenic 

lead having a Pbsor/Pb^ ratio, R\

GRAPHIC SOIUTIOW

Figure 2A is the graphic equivalent of the algebraic 

concordant age solution assuming a single period of 

past loss or gain of lead or uranium. The graphic 

equivalent of the correction for original radiogenic lead 

isjshown in figure 2B. A detailed discussion of the 

concordant graphical solutions for this case as well as 

for the other cases in this paper have been presented by 

Stieff and Stern (1961).

Samples A, B, and C (fig. 2A) are assumed to have 

been formed at the same time, t. Sample A has not 

been altered during its history, while B lost lead or 

gained uranium, and C lost uranium or gained lead at 

a time, tlt in the past. The Pb^/Pb206 ratio of the radiogenic lead present in A, B, and C at the time, t\, is 

given by the slope of the line, R*, passing through the 

points t and ti on the concordant age curve, times 

1/137.7, the ratio of the present day relative abundance 

of U235/U238. Thus, given R* and the discordant age 

data for one sample, B, it is possible to find the concordant unaltered age, t, and the time of alteration, ti,

by passing a line through the point B whose slope is 

R*y<. 137.7, and noting the two intersections with the 

concordant age curve.

Samples A, B, C and D (fig. 25) are also assumed to 

have been formed at the same time, t. At the time of 

their formation, however, B, C, and D were contaminated with different amounts of an older radiogenic 

lead having a Pb207/Pb206 ratio, R*. These samples 

would therefore, at the time of their formation, have 

the N207/N2 35 and N206 /N2 3& ratios shown by the points 

b, c, and d lying on the line of slope .5*X137.7, passing 

through the origin. If the samples have been unaltered 

since their formation, the N^j/Nzss and N^s/Nzss ratios 

will move from b, c, and d to B, C, and D. Given, for 

example, R* and the discordant N^/N^s and N^& / 

N238 age ratios for B, the concordant age, t, corrected 

for original radiogenic lead can be obtained by passing 

a line with a slope R*X 137.7 through the point B. In 

this instance, ti would represent the maximum age of 

the uraniferous source material which provides the 

older, contaminating radiogenic lead.
METHODS FOE EVALUATING DISCORDANT LEAD-ISOTOPE AGES Ell

COMMON Pb*<"/Pb*<>« RATIO GIVEN

The correction of lead-isotope-age data for contaminating common lead is an important problem 

closely related to the problem of correcting for original 

radiogenic lead. In some age calculations, the corrections for common lead are so large that it is necessary 

to know exactly the isotopic composition of the contaminating common lead. If the exact common-lead 

correction is not made, it is difficult to determine 

whether the lead-uranium and lead-lead discrepancies 

are a consequence of the assumed isotopic composition 

of the contaminating lead, experimental errors in the 

determination of the index isotope Pb204 or Pb208, or 

other factors.

One method of correcting for common lead, as 

indicated on page E8 is the use of the Pb204 abundance 

(and in certain special instances, the Pb208 abundance) 

as an "index" of the amount of common lead present. 

The isotopic composition of the contaminating common 

lead in the radioactive mineral may be assumed to be 

identical with the analyzed isotopic composition of 

the lead in associated nonradioactive minerals, such 

as galena. Alternately, if such data are not available, 

an isotopic composition may be chosen that is presumably representative of the lead available at the time 

of mineral formation. In either case, it is necessary 

to have very precise isotopic data on the abundance 

of the "index" isotope (Pb204 or Pb208 ) used in the 

correction calculations. The role of Pb204 as the 

"index" is further complicated by the fact that Pb204 

is the least abundant of the four isotopes and is consequently the most difficult to measure precisely in 

highly radiogenic samples. It would be desirable, 

therefore, if the common lead correction could be made 

without reference to either Pb204 or Pb208. For a 

calculation of this type to be valid, it is necessary that: 

(a) the sample be unaltered in any way since its formation, and (b) the sample be uncontaminated by original 

radiogenic lead.

An examination of equation 12 suggests an alternative solution. For ages in excess of current estimates 

of the age of the earth (4.5 billion years), R*, the ratio 

of the radiogenic Pb207 to Pb206, rapidly approaches the 

value of the Pb^/Pb206 ratio found in common leads. 

If the notation used in equation 13 is now modified 

so that We and W7 are proportional to the total number 

of atoms of Pb206 and Pb207 in a sample, then the number 

of atoms of common Pb206 and Pb207, N% and A7? will 

be given by the following equations:

n\

and

nl 2!

(14)

=N<7 (15)

Dividing equation 15 by equation 14 one gets,

+ '

RS nl

+-

2!

where R, a constant, =A75/A78=0.007262

Rc=NilNi, the ratio of the number of atoms of

Pb207 to Pb206 in the common lead. 

R&t WT/WS, the ratio of the total number of atoms 

of Pb207 to the number of atoms 

of U23S.

R»t= We/^Vg, the ratio of the total number of atoms 

of Pb206 to the number of atoms 

of U238.

Expanding and collecting terms, equation 16 takes 

the form

/^\I 

V nl nl 2! 2! /

(17)

For geologically possible values of Re, R5t, and R8t, 

equation 17 expanded to the third degree or higher has 

two positive real roots. The equations of odd degree 

also have one negative real root. The smaller positive 

root is the concordant age corrected for a common 

lead having a Pb207/Pb206 ratio, Re. The larger positive 

root generally has values in excess of current estimates 

of the age of the earth; these values are geologically 

impossible and consequently can be neglected. The 

negative real root does not have any geologic significance.

If computing facilities are not available, equation 

17 can be taken to the second power of t and the terms 

/i and/2 added. One then obtains

R =RC . (18)

Expanding equation 18 and solving for t,

(/ -~ 2a (19)

where

2a= (R«X? #M) 

b=(Rc\1 -R\2 ]

Equation 19 has one positive and one negative root 

for geologically possible values of Re, R&t, and R8t-
E12 SHORTER CONTRIBUTIONS TO GENERAL GEOLOGY

With approximately correct values for fi and /2 from 

table 2, the calculated age, t, from equation 19 will 

closely approach the exact concordant age, t, for the 

specific value of Rc, which is the Pb207/Pb206 ratio of 

the common lead contaminant.

EXAMPLE OF CALCULATION

In the frequent cases where chemical and isotopic data 

are available for only one uranium-bearing mineral from 

a given area, and where the Pb 207/Pb206 ratio of the 

contaminating common lead has been accurately determined, equations 17 and 19 can be used to calculate 

a concordant age that is corrected for common lead 

without using either Pb204 or Pb208 as the common-lead 

"index." In addition, the isotopic composition of the 

common lead required to give a concordant age may 

be calculated. This calculated isotopic composition 

may then be compared with the measured isotopic 

composition of lead extracted from associated nonradioactive minerals. The ages obtained by the use of 

equations 17 and 19 and the conventional "index" 

method may also be compared. This comparison requires that the quantitative data on the lead and 

uranium have small limits of error, the uranium-bearing 

mineral be fresh and unaltered, and that the Pb207 /Pb206 

ratio of the contaminating lead be known within relatively narrow limits. Failure of the two methods to 

agree probably indicates small errors in the isotopic 

analysis of the less abundant "index" isotope from 

either the lead extracted from the uranium-bearing 

material, the associated nonradioactive mineral, or 

both.

The following isotopic data are given for a hypothetical unaltered sample of uraninite, the true age of 

which is 950 million years, and which contains (chemical 

scale) 6.1237 percent Pb and 17.544 percent U.

Pb204

Isotopic composition of

lead from uraninite (in

atom percent)_________ 0. 86 

Isotopic composition of

associated common lead

(in atom percent)____ 1. 50g 23. 568 22. 655

The Pb207/Pb206 ratio of the common lead is 0.96126 .

52. 542 15. 983 30. 608

52. 26,

Using Pb 20* as the "index" of the amount of common 

lead present, the isotopic composition of the associated 

nonradiogenic lead, and correcting for the proportionate 

amounts of nonradiogenic Pb206, Pb207 and Pb 208, the 

remaining atom percent of radiogenic Pb206 and Pb207 

may be obtained as follows:

Pb2M pb2M Pb207 Pb208

Isotopic analysis of uraninite lead_ ______ __ _ _ _ 0. 8664 52. 542 15. 983 30. 608

Isotopic analysis of common lead. _ _______ 0. 8664 13. 53i 13. OOi 30. 010

Radiogenic lead produced

by uranium and thorium. 0. 000 39. Oli 2. 976 0. 598

The ratios and the discordant trial ages obtained 

from the age tables after correction for common lead 

are as follows:

Ratio Trial age (Millions of years)

N6/NS= 5g= __________ 0. 1583! 956 (Pb2°8/U288 age)

Nj/Ni =R& = _-_-__.- 1. 6629 1008 (Pb207/U285 age)

N,/Nt = _ _ _ __ _ __ _ _ 0. 07628 1122 (Pb20VPb206 age)

In this example, Re, the Pb^/Pb206 ratio of the contaminating common lead, was chosen to equal the 

Pb207/Pb206 ratio of the common lead used in the trial 

age calculation. Substituting also the ratios R5t, R&t, 

and R in equation 17 and solving the equations of the 

second to the tenth degree, we obtain the computed 

concordant ages given in table 4.

TABLE 4. Computed concordant ages obtained from equation 17

Degree of equation

2

3

^

5

Q

7

g

Q

10

Age corrected for common lead (millions of 

years)

945.58

948.49

949 79

950.06

950. 10

950.10

950 10

950.io

950 10

Second real root 

(millions of years)

-18705 .

+ 25085 .

8517 .7

6346 .4

562i.s

531 6 .2

5177 .6

5114.!

5085 .7

The smaller, positive real root of equation 17 rapidly 

converges on the hypothetical age, 950 m.y., corrected 

for common lead. In the calculation of the trial ages, 

an initial error of 2 percent was deliberately introduced 

into the Pb204 abundance of the hypothetical common 

lead used in the correction. This small error in the 

Pb204 abundance produced in turn errors of approximately 0.6, 6.0 and 18 percent in the trial Pb206/!!238, 

Pb207/u235 and Pb207/Pb206 ages, respectively. The age 

calculated using equation 17 did not include the index 

isotope and was not affected.

For equations of the third degree or higher, the larger 

real root is positive and converges more slowly on an 

age that is greater than the current estimates of the 

age of the earth. It should be noted that even for 

ages as much as 5,000 million years, equations of the 

tenth degree converge rapidly on the exact concordant 

age. This older age has no geological significance 

other than that a radiogenic lead with a Pb207/Pb206 

ratio equal to 0.9612 would have been produced in the 

time interval between 950 and 5,110 million years ago.

In order to use equation 19 it is necessary to express 

the analytical data as quantities proportional to the 

number of atoms present. Following the procedures 

given on page E8, the analytical data take the form 

^=0.053118
METHODS FOR EVALUATING DISCORDANT LEAD-ISOTOPE AGES E13

^8=7.3143 

W8 =1.5596 

'#7=0.4744!

£=0.007262

and the Pb207/Pb206 ratio of the contaminating common 

lead, Rc, is given as 0.96 126 .

For this approximate age range, it is also necessary 

to include the terms fi and /2 . The choice of an age 

ti from table 2 may be guided by the following generalizations on discordant ages related to uncertainties 

introduced by common lead corrections:

1. The Pb205/U238 age will be affected least by a poor 

choice of the isotopic composition for the commonlead correction.

2. The Pb207/!!235 and Pb207/Pb206 ages may lie above or 

below the Pb206/U238 age, depending on the isotopic 

composition of the common lead used in the 

correction.

3. The Pb207/Pb206 age will give the least satisfactory 

value for the actual age of the sample if the major 

source of error is confined to the common lead 

corrections.

On the basis of the trial age calculations, and using 

the generalizations mentioned above, values of j\ and 

/2 are chosen from table 2 for ^=950 m.y. Substituting the values for R5t, R8t, Re and R in equation 19 

one obtains:

2a=(Rc\l-R\2)

= 1.5843 XlO-2V2 

b=(Rc\i-R\2 ')

= 1.4068 X10-1V1 

c= (RR5t~RcR&t-\-R%-Rf2)

= -0.14079

Pb204 _ __--_-----_-_----_-------_

Pb208 _ _-_-__-_--_-----_--------_

Pb207 _ ___.._.__. ________________

Pb208 . __ . __ __ _____

Calculated

1. 480

23. 574

22. 661

52. 285

Measured

1. 5079

23. 568

22. 65*

52. 269

.=-

2a

(-1.4068 ±Vl.97909+0.4461i) t__- _- I xiU y

£=950 m.y.

The agreement between the age, t\, taken from table 

2, and the calculated age, t, obtained from equation 19, 

indicates that an additional age calculation with a new 

value for ,1 is not necessary.

649977 63 8

For certain geologic problems it may be useful tocompare the calculated isotopic composition of the 

contaminating common lead yielding a concordant age 

with the measured composition of the common lead 

used in the trial age calculation. Quantities proportional to the number of atoms of common Pb206 and 

Pb207, N% and N%, can be obtained from equation 14 

and 15, respectively. If the age tables are used, and 

£=950 m.y., N% is found to be 0.40962 and N$ is found 

to be 0.3937s. From the Pb208/Pb205 and Pb2M/Pb208 

ratios of the radiogenic lead, the quantities proportional 

to the number of atoms of Pb208 and Pb204 are found 

to be 0.9084s and 0.025716 . The calculated and the 

measured isotopic compositions (in atom percent) 

used hi the trial age calculation are as follows:

Isotopic composition

It does not follow from the differences in the Pb20i 

abundances given above that all the error must necessarily be restricted to the common lead. For the 

purpose of this example, the Pb2M/Pb208 ratio of the 

radiogenic lead was assumed to be correct. In actual 

practice, however, this measurement might be more 

suspect than the Pb2M/Pb208 ratio of the common-lead 

analysis. Under these conditions, differences in the 

calculated and "measured" index isotope may be used 

as evidence that the discordant age sequence reflects 

small errors in the determination of the index isotope 

in either the radiogenic lead, the common lead, or both.

GRAPHIC SOLUTION

The graphic equivalent of the algebraic solution of 

this case is shown in figure 3, where total tN2m/N2^ 

versus total W2o6/^V_38 ^s plotted instead of radiogenic 

NW/NZM versus radiogenic N2wlNm} as was done by 

Ahrens (1955a, b) and Wetherill (1956). Both the 

concordant and the discordant ages of a sample corrected for different amounts of a common lead with a 

specific Pb207/Pb206 ratio can now be determined from 

this new graph. The concordant age curve still represents the locus of those points whose NzmlNzx and 

-A^oe/AT.ss ratios give the same age, t. In this graph, the 

concordant age of an unaltered sample, A, corrected for 

a common lead with a specific Pb207/Pb206 ratio, R°, is 

obtained bypassing a line whose slope is equal to Re X 137.7 

through the point A, and noting the intersection of this 

line with the concordant age curve at the point,. It is
E14 SHORTER CONTRIBUTIONS TO GENERAL GEOLOGY

obvious that the age, t, is obtained without use of 

either Pb204 or Pb208 as the index of the common lead 

present.

This concordant age solution now may be compared 

graphically with the ages obtained by the more conventional "index" methods of correcting for contaminating common lead, which use either Pb204 or Pb208 

as the index isotope (fig. 3). First, the age is obtained 

from the ratio of the radiogenic Pb207/Pb206 using the 

appropriate index isotope, Pb204 or Pb208, and a common 

lead having the same Pb207/Pb206 ratio, Re . Then, 

with the aid of either the nomographs or the age tables 

that are available, a point, c, is obtained on the concordant age curve whose Nx»lNy& or N207/N235 ratio is 

equivalent in age to the radiogenic Pb207/Pb206 ratio age 

just calculated. A line is now drawn from the origin 

to this equivalent point, c, on the concordant age curve. 

The coordinate of the intersection, a, of the chord, Oc, 

and the line passing through A with the slope RC ~X. 137.7, 

will give the corresponding N207/N235 and Nm/N23s 

ratios that would be obtained by using the conventional 

index isotope method of common-lead correction. A 

comparison of the age equivalents of the coordinates of 

the intersections a and t may also help in a partial 

evaluation of the experimental errors encountered in the 

isotopic analysis of the generally far less abundant 

index isotopes.

TWO SAMPLES

AMOUNT OF CONTAMINATING RADIOGENIC T.TJAn AND ITS 

Pfo2°7/Pl>2o« RATIO UNKNOWN

From the preceding algebraic equations and from 

the graphic treatment (fig. 25) it can be seen that 

given analytical data for two different samples, an 

expression for t could be obtained without a knowledge 

of either the time and amount of alteration or amount 

of original radiogenic Pb206 and Pb207 present. This 

calculation can be made provided the two samples 

give different discordant ages and meet the following 

conditions:

1. The samples were deposited at the same time.

2. The samples were contaminated by an original 

radiogenic lead having the same Pb207/Pb206 ratio, or

3. The samples were altered at one time in the past.

4. The initial corrections for common lead have been 

correctly made, so that the remaining Pb206 and 

Pb207 represent only the total radiogenic lead in 

the samples.

5. The samples have not selectively lost or gained 

radioactive daughter products since the time of 

mineral formation.

6. The samples have not been recently altered.

These assumptions require, as a general rule, that 

if the discordant trial Pb206/U238 age of sample A is 

greater than the discordant trial Pb206/U238 age of

sample B, then the trial Pb^/Pb208 age of sample A 

must be equal to or greater than the trial Pb207/Pb206 

age of sample B. If this condition is not met, that is, 

the trial Pb^/Pb206 age of A is less than B while the 

trial Pb208/U238 age of A is greater than B, then the 

equations presented in this and in the following cases 

will probably yield one negative value of t. Such a 

pair of discordant ages may be interpreted as a failure 

of the two samples to fulfill one or more of the assumptions listed above.

Following the notation used in developing equation 

11, expressions may be written for samples A and B: 

For sample A, ^

R

nl ...+- 2!

-+A2

AM)"

1 V nl '"+*T+M (20)

and for'sample B

R

nl 21 -+x2 *

)'

<M!... +if!+v) (21)

where R5a= the ratio of the number of radiogenic Pb207 

atoms, NJa, to the number of U235 atoms, 

N5t>, now present in sample A. 

R8o=the ratio of the number of radiogenic 

Pb206 atoms, N6a, to the number of U238 

atoms, NSa, now present in sample A. 

R5J)=ihe ratio of the number of radiogenic Pb207 

atoms, N7 t,, to the number of U235 atoms, 

N5t> , now present in sample B. 

R86=the ratio of the number of radiogenic Pb206 

atoms, NM, to the number of U238 atoms, 

N8h, now present in sample B. 

Ai=the decay constant for U238 . 

\2 =the decay constant for U235 . 

R*} ,5*:= the unknown ratio of the number of atoms 

of original radiogenic Pb207 to Pb206, that is,

Accepting assumptions (1) to (6) above, equation 

20 is equated to equation 21. Expanding and collecting 

terms, the equation takes the form

p^i x? "1

I 2i\ 2i\ I

 R5a)]t

, R5t>R8a=Q. (22)
METHODS FOR EVALUATING DISCORDANT LEAD-ISOTOPE AGES E15

If computing facilities are not available, equation 22 

may be taken to the second degree and the terms /x 

and/2 added.

rx2 x2 1

2] (R&a #8&)+2] (^5& R&a)

Solving for t,

t ==-

2a (23)

where 2a=[\l(R5b -R5a) +\2 (RSa -

C = &a R& 5 6

For geologically possible values of R5a, R8a, RSI,, and 

RSJ> ; and with assumptions "l" through "4", given 

above, fulfilled; equations 22 and 23 have two positive 

real roots. Analogous to equations 12 and 13, the 

two positive roots of these equations approach the 

concordant ages corrected for original radiogenic lead, 

loss of lead, or gain of uranium.

EXAMPIE OF CAICUIATION

Equations 22 and 23 may be applied to two hypothetical, unaltered 500-million-year-old uraninite samples from the same deposit. Following the procedure 

used on page E8, the chemical and isotopic data for 

these two samples, A and B, are expressed after correction for common lead as follows:

Sample A Sample B

A/7 0 =0.11337

N5b= 0.1048s

JVa&=1.654,

A7"76 =0.1171o

Xi = 1.5309 x lO-'OyX2 = 9.7216 x 10- 10 yThe lead-uranium and lead-lead ratios and trial leaduranium and lead-lead ages of samples A and B using 

the age tables corrected only for original common 

lead are:

Sample A Patio 

0.097794 

0.87925 

0.065290

Sample B

AWfl86=fi86= -- 0.11460

NiblNsb=R6b= - 1.116s 

N7b/N6b= --------- 0.070772

Trial age 

(million years)

608 (Pb206 /U238 age) 

648 (Pb207 /U235 age) 

798 (Pb20VPb206 age)

706 (Pb206 /U233 age) 

772 (Pb207 /U235 age) 

968 (Pb207 /Pb206 age)

Substituting the values for R5a, Rsa, RSJ> and RSD in 

equation 22 and solving the equations of the second 

to the tenth degree, the ages given in table 5 are 

obtained. It can be seen that in the age range of 

500jnillion years, ages corrected for original radiogenic

TABLE 5. Computed concordant ages obtained from equation 22

Degree of equation

2___ __ _ _ ____ _

3-____---_____-----.__-

4 _ _ _ _ __ ___ _

5__-__-_--______--___--

6__ _____ _ _ ___ _ _

7

8_____ _ __- 

9__ _ _________________

10__ __ ________________

Age (millions of years)

Corrected for original 

radiogenic lead

477.09 

496.72 

499.64 

499.93 

499.95 

499.96 

499.96 

499.66 

499.96

Corrected for loss or 

gain of lead or uranium

34502 

164s.8 

144i.2 

1400 .9 

13927

139!.!

I39o.9 

I39o.g 

139o.8

lead will fall within 0.1 percent of the exact concordant 

age when an equation of the fourth degree is used. 

The "lead-loss" age for the fourth-degree equation 

will, in contrast, be within 3.7 percent of the exact 

concordant "lead-loss" age. An equation of the sixth 

degree would be required to reduce the difference 

between the computed and exact "lead-loss" age to 

less than 0.1 percent. For most geologic problems 

falling within this age range, the two concordant ages 

obtained from a fifth- or sixth-degree equation would 

be more than adequate.

In order to use equation 23, it is first necessary to 

examine the values of /i and/2 in table 2 for a time, ti, 

of around 600 million years. This examination suggests that the terms including both /x and /2 should be 

used in the calculation. As an age including an additional correction for original radiogenic lead must necessarily be less than the lowest Pb206/U238 age in the group 

of samples corrected only for common lead, initial values for /i and/2 equivalent to an age of ^=550 m.y. are 

chosen for the calculation. Substituting the following 

in equation 23,

£56=1.1168 /!=0.00010 

#8a=0.097794 #86=0.11460 /2 =0.02928

where

a

5 ) + 94.510 (-0.01681 )]X10-2V1 

= -1.02758 X10-2V2

= i 51) SO 2 2 (R8a8a SJ>

= [1.5369 (0.23755 +9.7216 (0.01681 )]X10-1V1

=0.20167 X10~1V1

C = [R5aR& D R& bRsa +/1 (Rs & R&a) +/2 (Rsa R& &) ]

0.109216+0.000024 -0.000492]

= 0.008922
E16 SHORTER CONTRIBUTIONS TO GENERAL GEOLOGY

t becomes,

£=-

2a

-0.20167 ±V(.20167) 2 -2X -1.027 58 X-0.00892

=L 1.0275g 

-0.2016+0.14945

1.02758

X1010t

io10?/

 1.027 58

=0.05082 X101(V

=508 m.y.

From the fact that the calculated age, £=508 m.y., 

included values from table 2 of/x and/2 for ^=550 m.y., 

it is obvious that a recalculation of t using values of 

/! and/2 for t slightly less than 508 m.y. would be better. 

There is, however, an exceedingly sensitive test of the 

departure of the age, £=508 m.y., from the exact solution of equation 23. This test consists of comparing 

the calculated ratios of the original radiogenic leads 

in samples A and B (Ra and Rfr,) using equations 20 

and 21 and £=0.0508X101(V- As a first approximation, 

the values of /x and/2 in equations 20 and 21 may be 

chosen from table 2 for ^=510 m.y. A simpler procedure is to substitute the ratios N6/N8 and N7/N5 

obtained from the age table for £=508 m.y. for the

terms /t+ *, +\t and /2+^-|^-+X2 £ inequation

20 and 21. Using the age tables, and are

respectively, 0.1051, and 0.1040.

If the geologic requirements of the age problem are 

not stringent, and the disagreement between R* and R%, 

is acceptable, the calculated age may be rounded down 

to the nearest five million years and reported as £=505 

m.y. If, however, further refinements are required by 

the nature of the geologic problem, £ may be recalculated using equation 23. The values of /i and /a, 

0.000079 and 0.0224OS, for £i=505 m.y., are obtained from 

table 2 by linear extrapolation. The recalculated age 

of samples A and B using equation 23 then becomes, 

£=500.06 m.y.

A test of this calculated age at £=501 m.y., using 

equations 20 and 21 gives for R* and R% respectively, 

0.10297 and 0.1028X . The condition ^*==^=0.10266 

occurs at £=500 m.y. For almost any conceivable 

problem, this very close agreement between IJ*a and R% 

would be more than adequate. As the second recalculated age is still less than the age ti used in choosing 

Ji and/2 , the final age for samples A and B would now 

be rounded down to the nearest 5 m.y. and reported as, 

£=500 m.y.

In areas where detailed age studies are being made on 

a suite of uranium ores, isotopic data may also have been 

obtained for the lead in the associated nonradioactive 

minerals. If the lead in these associated minerals is 

abnormally enriched in Pb206 and Pb207 relative to 

Pb204, the Pb207/Pb206 ratio of this radiogenically enriched lead may now be compared with the calculated 

ratio of the original radiogenic lead, R*} required to 

produce concordant lead-uranium and lead-lead ages in 

any particular pair of ore samples. Agreement between the calculated ratio of the required original 

radiogenic lead and the measured Pb207/Pb206 ratio of 

the radiogenic lead in the associated nonradioactive 

minerals, could be used as convincing evidence in the 

evaluation of a discordant age pattern in terms of the 

presence of original radiogenic lead. Evidence of 

alteration, conversely, could be used to support a 

"lead-loss" type of evaluation.

The complexity of most geologic age problems as well 

as an awareness of the experimental uncertainties 

usually suggest the desirability of studying, where 

possible, more than two samples from the same area. 

Similarities in the calculated concordant ages and the 

R* obtained from two or more pairs would not only 

support the interpretation of the discordant age data in 

terms of a specific process, but also further limit 

alternative interpretations.

GRAPHIC SOLUTION

The graphic equivalent of the algebraic concordant 

age solution for two samples is shown in figure 2B. 

The concordant age corrected for the presence of an 

older generation of radiogenic lead can be obtained by 

passing a line through the two points, B and C, and 

noting the intersection, £, with the concordant age 

curve having the smallest N2o6/N23& and N^/N^a ratios. 

The slope of this straight line divided by 137.7 will 

equal R*, the Pb207/Pb206 ratio of the older radiogenic 

lead. The intersection giving the higher age value, 

ti, corresponds to the age obtained after correcting for 

the gain or losses of lead or uranium as previously 

described. This graph makes it clear, as do the 

equations, that the two concordant age solutions £ and 

ti are mathematically equally acceptable. The choice 

of either t or £t as the most probable concordant age 

for both B and C must be based on other evidence.

AMOUNT OF CONTAMINATINQ COMMON LEAD AND ITS 

Pfo2°7/pb2<w RATIO UNKNOWN

Following the arguments used in the previous 

example and the notation used in the correction for 

common lead, a concordant age, based on the following 

assumptions, can be found for a pair of samples giving 

different discordant trial ages. 

1. The samples were deposited at the same time.
METHODS FOR EVALUATING DISCORDANT LEAD-ISOTOPE AGES E17

The samples were contaminated only by a common

lead having the same Pb207/Pb206 ratio. 

The samples have not been altered since deposition.

Sample A, equation 16, takes the form,

E

...+- 2 ,

T> /(^lO" I (^lO I \ .\ tts,a-( r - ~> 2l *"Xlv

and sample B.

(24)

R

5i n\ 2! h 2

st *~

(25)

where N^JN^a=Rac , the ratio of the number of atoms of 

Pb207 to Pb208 in the contaminating 

common lead (sample A).

N£jNw=Rt, the ratio of the number of atoms of 

Pb207 to Pb206 in the contaminating 

common lead (sample B] .

^5t<z=the ratio of the total number of atoms 

of Pb207 , W7a , to the number of U235 

atoms, N5a , now present in sample A.

.R8 ta=the ratio of the total number of atoms 

of Pb206, W6a , to the number of U238 

atoms, NSa, now present in sample A.

-S5 « 6=the ratio of the total number of atoms 

of Pb207, W76, to the number of U235 

atoms, NM, now present in sample B.

S8Z6=the ratio of the total number of atoms

of Pb206, W66, to the number of U238

atoms, NSD, now present in sample B.

Equating equations 24 and 25, (assumption 2)

clearing and collecting terms,

\n

^ (R^tD (R&ta

-I 

I

TM (RM-R^+** (R8ta -R8t!H I2! 2! |

(26)

If computing facilities are not available, equation 

26 may be expanded to the second degree and the 

terms/i and/2 added. Solving this equation for t one 

obtains:

t=

-b± V62 - 

2a (27)

where

8 ta R& ift)

C = R5 taR& lit & la 5 ta & la

For geologically possible values of R5ta, R8ta, -R5i6 , 

and R8ti> fulfilling assumption 1, 2, and 3 equations of 

degree 3 or greater, equation 26 has two positive, real 

roots. The smaller root, as was the case for equation 

17, approaches the concordant age corrected for a 

common lead whose Pb207/Pb208 ratio, Rc , is given by 

equation 24 or 25. Equation 27 has one positive and 

one negative root. The positive root for approximately correct values of /i and /2 approaches the 

smaller root of equation 26.

EXAMPLE OF CALCULATION

An example of the use of equation 26 in the correction 

for common lead is given below. The following isotopic and chemical data are given for two hypothetical, 

unaltered 950-million-year-old uraninite samples, A and 

B from a single deposit.

Sample

A.... ______

£__._______

Isotopic composition (in atom percent)

Pb204

0. 8664 

0. 6523

Pb206

52. 542 

62. 647

Pbao?

15. 983 

13. 654

Pb20a

30. 608 

23. 047

Uranium and lead 

content (chemical 

scale)

Percent 

U

17. 544 

30. 7S3

Percent 

Pb

6. 1237 

7. 94«

Assuming an isotopic composition for the common 

lead contamination in samples A and B to be in atom 

percent,

. 4852 23. 764 22. 278 52. 473

the following discordant trial age sequences are obtained 

using the age tables and Pb204 as the index isotope:

Sample A. 

Sample B.

948 m.y. 1, 010 m.y. 1, 148 m.y. 

950 m.y. 984 m.y. 1, 064 m.y.

These discordant trial ages may be compared with the 

concordant ages obtained from equation 26 given in 

table 6. The smaller root of equation 26 converges 

on the concordant age corrected for common lead even 

though the isotopic composition of the contaminating 

common lead does not appear in the calculation. The 

similarity of the trial pb206/U238 age to the concordant 

age corrected for common lead might be used as one
E18 SHORTER CONTRIBUTIONS TO GENERAL GEOLOGY

argument that the discordant trial age sequence was 

the result of an incorrect assumption as to the isotopic 

composition of the contaminating common lead.

TABLE 6. Computed concordant ages obtained from equation 26

Degree of equation

2 ____ ._..__________..

3. .__..____..___. .__

4 ___. ___ __

5__ __..____.._________.

6. _.._.__._______ ___

7__ ..______._._._..____

8______. _______________

9____________________

10.. -._.________. _

Concordant age corrected for common lead 

(millions of years)

945.42

948.33

949. 83

949.90

949. 94

949. 94

949.94

949. 94

949. 94

Second real root 

(millions of years)

25068.

8515. 4

6345.,

56204

5315 .4

5178. 9

5113.4

508s.!

Using the generalizations given on page E8 as a guide 

in the evaluation of the discordant trial ages of samples 

A and B, an initial choice of ^=950 m.y., corresponding 

to the Pb2o6/U208 age, is made in selecting the constants 

/i and/2 from table 2. Following the procedure given 

on page E8, the chemical and isotopic data for samples 

A and B are expressed below as quantities proportional 

to the total number of atoms of lead and uranium in the 

sample at the time of analysis, and as ratios of the total 

number of atoms of Pb206 and Pb207 to U238 and U235 .

Sample A

N5a= 0.053 118 

#8a=7.3143 

'#6a=1.5598

'#7a =0.4744i

_ 

0.4744o " 8lB ~

0.05312i 

1.5598 __

Sample B

#56=0.093108

#86=12.82!

'#66=2.4164

'_V76 -=0.52665 

0.5266s _=5.b5O4

for *i = 950 m.y. 

Xi=1.5369 X10-10;rl 

X2 =9.7218 X10-1V-1

0.093108 

_2.4164

-i2T8T2 -°-18847

/i=0.00054 

/2 =0.16819

Substituting the values given above in equation 27:

t==-

2a

where

5.3967 X10~2y1

(_S6 1{> R5ta) -f- X2 (RS ta~ R&tb)

=0.47960

K4

l>~~R&to) ~T~J2

7922 ±V(4.7922) 2 -2X-5.3967X0.47967j 

-5.3967 /

=A.7922 ± V28.1417\y T nio 

V -5.3967 /

/:4.7922 -5.3049

-5.3967 

= 0.09500 X1010 years=950 m.y.

The agreement between the calculated value of t, 

950 m.y., (table 6) and the value of ti used in the 

choice of j\ and /2 from table 2 indicate that a recalculation of t using slightly different values of fi and /2 

is unnecessary. The calculated values of the Pb207/ 

Pb206 ratios for the common lead (Rca and R\ in samples 

A and B} using 2=950 m.y., the age tables, and equations 24 and 25, are respectively, 0.96132 and 0.9611e; 

the average calculated value is 0.96125. The Pb207/Pb206 

ratio used in setting up this hypothetical example was 

0.96127 .

The quantity proportional to the number of atoms 

of common Pb206, in sample A is equal to 0.40964 and 

can be derived from equation 14 by substituting for 

t, 0.950X10 10 y. The proportional number of atoms of 

common Pb207 is obtained using the ratio NlaIN$a= 

0.96125, and is equal to 0.39377 . Using the Pb206/Pb20* 

and the Pb206/Pb208 ratios obtained from the isotopic 

analysis of sample A, quantities proportional to the 

total number of atoms of Pb204 and Pb208 are calculated 

to be 0.02572 and 0.90853 . The actual isotopic composition used in setting up this example, the atom-percent 

abundances of the original common lead calculated from 

the number of atoms of Pb204, Pb206, Pb207 and Pb208, 

and the isotopic composition assumed to be present in 

the trial age calculations are given below:

Isotopic composition in atom percent

Isotope

Pb204_ _ _ . __ _ __ _--

Pb206_ _ - _--- -

Pb207 ._ --__ - _

Pb208 .- __ __-____..__

Actual

1. 480

23. 574

22. 66!

52. 284

Calculated

1. 480

23. 575

22. 66!

52. 284

Assumed

1.48s

23. 764

22. 278

52. 473

Thus, the ages obtained from equations 26 or 27 

permit the calculation of a concordant age, t, without 

the use of either Pb204 or Pb208 as the index isotope, and 

without a knowledge of the isotopic composition of the 

contaminating common lead. Alternatively, a compar-
METHODS FOR EVALUATING DISCORDANT LEAD-ISOTOPE AGES E19

ison of the ages obtained from these equations with the 

age obtained by using the conventional methods of 

common lead correction and age calculation permits an 

evaluation of the errors that may be introduced in the 

correction for common lead. If the calculated isotopic 

composition of the common lead necessary to bring the 

lead-uranium ages of samples A and B into agreement 

seems unreasonable, additional field and laboratory 

evidence may then be sought to determine if the disdiscordant trial age calculations were produced by 

experimental errors, processes of alteration, or by the 

presence of original radiogenic lead.

GRAPHIC SOLUTION

The graphic equivalent of the algebraic solution for 

the above example is shown hi figure 3, a plot of the

FIGURE 3. Ratios of the total number of atoms of Pb*» to U»s, 'NwtNsu, plotted 

against the ratio of the total number of atoms of Pb*" to U**8, 'NiotlNm. Unal. 

tered uranium-bearing samples contaminated with a common lead having a

ratio of the total number of atoms of Pb207 to U235, 

Waw/TVas versus the total number of atoms of Pb206 to 

U238, W206/U238. The concordant age is obtained by 

passing a line through the coordinates of the points A 

and B (fig. 3) and observing the intersection of this line 

with the concordant age curve. The slope of this line 

divided by 137.7 will be the Pb207/Pb206 ratio, Re, of the

contaminating common lead which will give the same 

concordant age for both samples, A and B. This ratio 

may now be directly compared with the experimentally 

determined Pb207/Pb206 ratio in related nonradioactive 

minerals, such as galena, pyrite, or feldspar. Failure 

of the experimentally and graphically determined Pb207/ 

Pb206 ratios to agree, or an abnormal graphic value for 

Rc, will indicate a more complex geologic history than 

was initially assumed. As has been mentioned, the 

same age can be obtained graphically from plots of 

N2Q7/N2M versus N2Qt>/N2Q4 (fig. I A) even though the 

samples have been recently altered.

AMOUNT OF CONTAMINATINGIKADIOGBNIC AND COMMON 

LEAD AND THE PD^ifPDm RATIO UNKNOWN

In the discussion of figure lA it was noted that in 

plots of the Pb207/Pb204 versus Pb206/Pb204 ratios (and 

in the case where thorium is absent, Pb207/Pb208 versus 

Pb208/Pb208 ), two or more different mixtures of a single 

common lead and a single radiogenic lead would lie 

on the same straight line. The slope, R*, of this 

line is the Pb207/Pb206 ratio of the radiogenic lead, and 

on this line will lie the Pb206 /Pb204 and Pb207/Pb204 (or 

the Pb^/Pb208 and Pb207/Pb208 ) ratios of the contaminating common lead. This graphical treatment and 

the difference plots in figure 1J5 suggest that a useful 

modification may be obtained if the chemical and 

isotopic data for two or more samples were expressed 

as ratios of the total numbers of atoms of Pb206, Pb207, 

U235, and U238 to the total number of Pb204 (or Pb208) 

atoms. Using this change in the notation of equations 

11, 12, and 17, it is possible to calculate a concordant 

geologic age from the discordant age data without 

knowledge of the amount or Pb207/Pb206 ratio of the 

contaminating common lead. In this case, however, 

the ratio of the contaminating radiogenic lead must 

be known.

For the above calculation to be valid, it would be 

necessary (as in the previous cases), for the two or 

more samples in question to yield different discordant 

ages and to meet conditions (1) through (4), and 

either (5) or (6) given below.

(1) The samples were formed at essentially the same 

time.

(2) The samples have not been recently altered.

(3) The samples at the time of their formation contamed a common lead with the same Pb207/Pb206 

ratio.

(4) The samples could not contain thorium if Pb208 is 

used as the index isotope.

(5) The samples at the time of then1 formation contained an original radiogenic lead with the same 

Pb^/Pb206 ratio, or
E20 SHORTER CONTRIBUTIONS TO GENERAL GEOLOGY

(6) The samples were altered only once and at the 

same time in the history of the material.

This calculation does involve the precise determination of the abundance of the index isotope, Pb204 (or 

Pb208). The relative simplicity of this modification, 

however, and the many instances where it can be used 

to advantage compensate, in part, for this obvious 

shortcoming.

Following the notation used in equations 9 and 10, 

one can write for sample A

and

where

=.^ (29)

 ^6*= a quantity proportional to the sum of the 

number of atoms of common and original 

radiogenic Pb206 in sample A.

 ^7*= & quantity proportional to the sum of the 

number of atoms of common and original 

radiogenic Pb207 in sample A.

If both sides of equations 28 and 29 are now divided 

by the index isotope, Wia, a quantity proportional to 

the total number of either Pb204or Pb208 atoms in 

sample A, depending on the choice of the index isotope, 

one obtains the following:

and

where

=WS (30)

=W?* (31)

Similarly, 

ratios of the

and

where

equations for sample B, expressed as 

index isotope, N^, are:

(32)

n\ ' ' ' ' 2! (33)

"# =* 

"N*=

ff\Tc*__

In the graphic treatment of the isotopic data for a 

mixture of a common and a radiogenic lead expressed 

as ratios of Pb204 or Pb 208 (fig. IB), it has been mentioned 

that the slope of the line passing through the origin 

and the points C B is equal to the ratio of the added 

radiogenic lead, R*. The point C B in figure IB is 

equivalent to a point having the coordinates('Nfa 'Nil) 

and (We* 'AO- Therefore, subtracting equation 33 

from equation 31 and dividing by equation 32 subtracted 

from equation 30 one obtains:

...+' 2!

>-(W&z -W86)(^-"...+^

or

£ n\ ...+- 2!

'£8(a_6) /(M)« ,(XiO*, wM ~(~ti~ - - +~2r+XivJ

where the normalized difference ratios are

=R*

i=_ / Arc*

(34)

and

'N*-'i 

''N^-'l ^=0.

/ Ajc*_ / AT"** AT*

75*__ -^V7g -^V7ft -^V7fl ^ ~/A7c*_/A7'c*~A7'*

and where the same index isotope (Pb204 or 

in both samples A and B.

Clearing and collecting terms

^^i-^MY* /fl*xf _-gxi^a

V n! w! / " ""A 2! 2! >

used

_6) =0. (35)

If computing facilities are not available, equation 35 

may be carried only to the second power of t and the 

terms j\ and /2 added. Solving this equation for t one 

obtains,

t=-

2a (36)

where

c=
METHODS FOR EVALUATING DISCORDANT LEAD-ISOTOPE AGES E21

For geologically possible values of R*, rR5 (a-v, and 

'#8(a-&), and fulfilling the assumptions (1) through (6), 

equations 35 and 36 have two positive roots. If 

original radiogenic lead is assumed to be responsible 

for the discordant trial ages obtained on sample A and 

B, the smaller root will approach, as the degree of the 

equation increases, a concordant age corrected both for 

the presence of an original radiogenic lead having a 

Pb207/Pb206 ratio, R*, and a single common lead. The 

isotopic composition of the common lead need not be 

known. If past alteration is assumed to be responsible 

for the discordant trial ages, the larger root will approach the concordant "lead-loss" age as the degree 

of the equation increases. For the latter assumption, 

the smaller root will approach the time in the past when 

the alteration took place, and R* will become the Pb207/ 

Pb206 ratio of the radiogenic lead produced by the uranium in the time interval, (t ti), given by the two roots.

EXAMPLE OF CALCULATION

An example of the use of both equations 35 and 36 

in the calculation of a concordant age, t, corrected for 

unknown amounts and ratios of both a common and 

an original radiogenic lead are given below. In this 

example, the absence of Th232 in the chemical analyses 

suggests the use of the much more abundant isotope, 

Pb208, rather than Pb204 as the index of the common lead 

present. The isotopic and chemical data for two 

hypothetical unaltered 1,400-million-year-old uraninites, A and B, from the same deposit are:

Sample

A......

£______

Isotopic composition (in atom percent)

Pbaw

0. 2107 

0. 07646

Pb2«

80. 697

87. 684

Pb207

11. 80j 

9. 5945

Pb208

7. 2889 

2. 6448

Uranium and lead 

content (chemical 

scale)

Percent

U

31. 550 

53. 17!

Percent 

Pb

9. 7434 

13. 412

Assuming a common lead whose isotopic composition is Pb204= 1.52s, Pb206=22.867, Pb207 =22.869, and 

Pb208=52.753 , trial age calculations yield the following 

discordant results:

Sample A_________ 1,605 m.y.

Sampled______ 1,480 m.y.

1,715 m.y. 

1,530 m.y.

1,850 m.y. 

1,650 m.y.

Using the data given above for samples A and B in 

equation 35, the concordant ages were obtained for 

equations of the second to the tenth degree (table 7). 

The smaller positive roots yield concordant ages 

corrected for original radiogenic lead given a Pb207/Pb206 

ratio, R*, of 0.25004 . The larger root of the tenth649977 63 4

degree equation is very close to the concordant age of 

samples A and B corrected for loss or gain of lead or 

uranium during a single period of alteration 1,400 

million years ago.

TABLE 7. Computed concordant ages obtained from equation 35

Degree of equation

2

3

4

5

7

g

g

10

Age (millions of years)

Corrected for original 

radiogenic lead

1266 .7 

1344 . 2

1383 .5

1396 . 3 

1399 .4 

140o.0 

1400.! 

140o.2 

140o.2

Corrected for loss or 

gain of lead or uranium

64266 . 

4648 .5. 

3120 .i 

276! .0 

2652 .7

26oM 

2606S 

2605 ;ft

Following the procedure and notation used^on page 

E8, the chemical and isotopic data for samples A and B 

are expressed in terms proportional to the total number 

of atoms present.

Sample A Sample B

#5a = 0.095526 #56=0.16099

#8o=13.154 #85 =22.169

W6a=3.8296

'#7a=0.56004

W08 « = 0.3459!

In the above tabulation, *-V0s_ and W08& are proportional to the total number of Pb208 index atoms in 

samples A and B at the present time.

Expressing these data as multiples of the index

_Vr isotope, Pb208, that is, 'N5a==pr^-) and so on, one obtains:

Sample A 

' #5a=0.27616 

' #8o =38.027 

'Wte=11.07i 

'«_\r7 «=1.619o

and for £1= 1,350 m.y.

_/i=0.00157 

where

'08a

Sample B 

' #56 = 0.93085

'#8 6=128.18

"#65 =33.154

/2 =0.54150

^3.0g8,l (£*=0.25004 , given)

6 , (£=0.007262)

Substituting in equation 36

C=(R'R5(a-J>) R*'Rs(t

and solving for t,
E22 SHORTER CONTRIBUTIONS TO GENERAL GEOLOGY

t'

2a

-0 0.31369 ±VOT098401 -2X-0.09571 X-0.04250A

-0.0957 X

XIO10?/

= 1,384 m.y.

The calculated value for t, 1,384, lies above the initial 

choice of t\, 1,350, used to obtain j\ and/2 from table 2. 

This fact indicates that in using table 2, a choice for t 

greater than 1,384 and less than 1,480 should be made 

for the second age calculation if, for geologic reasons, 

the first age calculated is not acceptable.

A very sensitive test of the departure of the value 

#=1,384 m.y. from the concordant age for samples A 

and B may be made by comparing the calculated value 

of R* (a -v using equation 34 and £=0.1384X10 10;?/ 

with the observed or assumed value for R* used in 

equation 36. The value of R* calculated in this way 

and the given R* are, respectively, 0.2083 and 0.25004 . 

For the value 2=1,400 m.y., the calculated and given 

values of R* are, respectively, 0.24985 and 2,5004 . As 

can be seen, relatively small changes in t will result in 

substantial changes in the calculated R*.

GRAPHIC SOLUTION

The graphic equivalent of the algebraic solution for 

the above example is shown in figure 4.

If two unaltered samples, A and B, containing different amounts of lead and uranium were formed at the 

same time and were contaminated only with a common 

lead of uniform Pb 207/Pb 206 ratio, Rc , the point (A B) 

whose coordinates are given by the normalized difference ratios

(N2S5/N20l)A-

(N258fN20i)Awould lie on the concordant age curve.

The problem of the three component lead mixtures, 

Rc, R*, and R can now be considered (fig. 4). Assume 

that in addition to a common lead having the same 

Pb207/Pb206 ratio Rc, samples A and B have received 

different amounts of an older generation of radiogenic 

lead with a Pb207/Pb206 ratio, R*. Then the normalized 

difference ratios of these two component mixtures, Rc 

and R* (see figure IB) would lie on a line passing through 

the origin whose slope is equal to JR*X137.7; for 

example, the point (c d). The amounts of radiogenic 

Pb206 and Pb207 produced in the time interval Q-t may

FIGUBE 4. Normalized difference ratios of 

(Nm/NiM)x- for three pairs of uranium-bearing samples which were

contaminated by the same common lead and) which, in addition, were altered at a 

time, t, in the past or contaminated by an older radiogenic lead, R*.

now be added graphically to the point (c d) by 

passing a line through the point (c d) with the slope, R, 

and of a length equal to that of the line segment from 

the origin to the point (A B). The coordinates of the 

point (C D) thus represent the discordant ages that 

are obtained when the normalized difference ratios are 

plotted for a pair of samples each containing three 

component lead mixtures.

THREE SAMPLES

AMOUNT AND Pb207/pfc206 RATIOS OF CONTAMINATING 

COMMON AND RADIOGENIC TJ3AD UNKNOWN

From the argument presented above it is clear that 

if isotopic and chemical data are available for three 

samples fulfilling the conditions set forth, for the two 

sample cases it would be possible to calculate a concordant age without knowledge of either the amounts or 

isotopic ratios of the contaminating common and 

original radiogenic lead. Using equation 34, the 

following equations can be written for the two pairs 

of samples (A and B, and A and C) where the isotopic 

and quantitative lead and uranium data are expressed
METHODS FOR EVALUATING DISCORDANT LEAD-ISOTOPE AGES E23

as normalized difference ratios of the number of atoms 

of lead and uranium to the number of atoms of the index isotope, Pb201 or Pb208 : 

Sample pair A-B,

R

jft Kr/i 6i " ~

2 0tt

2!

'jRsca-6) (

M."

nl 2!

=R* (37)

where the normalized difference ratios

I Tt _ ^

-"-8(0-6) 

and sample pair A-C,

£ =R* (38)

where

Placing equation 37 equal to equation 38 and collecting terms, one obtains:

rxn \n ~i

n\ (/'^'6 <a-c>"~"'^s < a- &))~^^J C-^8(a-6) '#8(8-0) M» 

rx2 x2 I

2f (/-^5(a-c)~ /^5(a-6)) +2|" ('^(a-ft) '^8(0-0) \t2

"t" -f^5(a-6) R&(a-c) ~~ Rs(a-c) R&(a-l)) Q- (39)

If computing facilities are not available, equation 

39 may be taken only to the second degree and the 

terms fi and /2 added. Solving for t, one obtains:

f = -

2a (40)

where

Equation 39 taken to the third degree or higher has 

two positive roots for values of the ratios, 'R5(a^)t

'jR8{0_ 6) , 'jRsca-o, and 'jRsca-o, that are geologically significant. As in the previous case, with an increase in 

the degree of the equations, these roots approach the 

exact concordant age corrected for original radiogenic 

lead or loss of lead. For most geologically significant 

values for the ratios, equation 40 will also have two 

positive roots. For values of ti less than the lowest 

trial lead-uranium age, the smaller root will approach 

rapidly the exact concordant age corrected for original 

radiogenic lead. With the same value for ti, the larger 

root has no geologic meaning. To obtain the "leadloss" age, different values of ti equal to or greater 

than the largest trial lead-lead age must be substituted. 

The initial choice of ti may be guided by a rough 

graphical solution of

versus

EXAMPLE OF CALCULATION

xIf the data for the third sample, C, are added to the 

age data given on page E21, it is possible to use equations 

39 and 40 to calculate a concordant age, t, without 

having any knowledge of the amounts or ratios of the 

contaminating original radiogenic and common lead.

Sample

C. .......

Isotopic composition (in atom percent)

Pb204

0. 2263

Pb206

80. 657

Pt)207

11. 288

Pt)208

7. 827

Uranium and lead content (chemical scale)

Percent U

46. 84?

Percent 

Pb

13. 613

Assuming a common lead whose isotopic composition

is Pb204 =1.525 , Pb206=22.867, Pb207=22.869 , and 

Pb20s =52.753 , trial age calculations yield the following 

discordant results:

Sample C_

Pb206/TJ238

1,515 m.y. 1,590 m.y. 1,690 m.y.

If the data for samples A and B given in the preceding 

example (p. E21) are combined with the data for sample 

C, equation 39 can now be used for concordant age 

calculations provided that the assumptions given on 

page E20 can be fulfilled. Table 8 shows the results of 

these calculations for equations of the second to the 

tenth degree. These calculated ages are in close 

agreement with the ages given for samples A and B 

in table 7.
E24 SHORTER CONTRIBUTIONS TO GENERAL GEOLOGY

TABLE 8. Computed concordant ages obtained from equation 39

Degree of equation

2__

3___. __________________

4_ _ _ _

5______________________

6____ ___ _

7__ _ _

8______________________

9____ __._ _ _ _

10

Age (millions of years)

Corrected for original 

radiogenic lead

12670

1334 .4 

1383 .5

1396 .4 

1399 .5 

1400 .i 

140o.2 

1400 .2 

140o.2

Corrected for loss or 

gain of lead or 

uranium

64705 . 

4653 . 5 

3122 . 7 

27630 

2654 .6 

2620 .9 

261o.8 

2608 i 

2607 .4

If the analytical data for the three samples are 

expressed as multiples of the number of Pb208 atoms in 

samples A, B, and C, respectively, one obtains,

Sample B

"Ar66 =33.154 

"N7b= 3.627:

Sample A 

'Ar5o=0.27616 

'^=38.027 

"JVta =11.07i

'W7«=1.619o

then,

/J?5(a_ 6) = 3.0683 

'J?5 (a_ C)=62.883

and for . x = 1,350, 

/1 =0.0015

Solving equation 40 for t,

Sample C 

'N5e=0.2733s

"N1e= 1.442s 

/^8(a- &) =0.24498

/2 =0.54150

2a

where

2a=[\i('R5(a - c) '^5(0-

= -22.623 X10-2V1

0=[Xi( Rs(a-c) '^5(0-

=75.068 X10- 107/- 1

= -10.18.

I" i

_J -75.068 ±V(75.068) 2 -2X-22.623 X-10TT8^

 22.62,

xio1(v

-75.068 ±V5174.6 

-22.623

-75.068+71.934

X101(V

 22.623 

= 1,385 m.y.

X101(V=0.1385X101(V

The small discrepancy in the calculated age obtained 

from equations 36 and 40 is due to the fact that samples 

A and G are quite similar in isotopic composition. As 

a result, the difference between 'N^a 'N^c is very 

small, 0.00278 . This small number appears as the 

denominator of the term E5(a-o and consequently 

affects the final age calculation.

The extent of the departure of equation 40 from the 

concordant age resulting from the use of values /i and 

/2 for ^ = 1,350 m.y. (from table 2) can be determined 

by comparing #*«,_&) and #*(&_ c) . Using equations 

37 and 38 and £=.1385X1010 y, #V&) and #*<B-c> are 

respectively 0.21593 and 0.2333. At . = 1,400 m.y., 

7?* (a_ 6) and R*(a-o are 0.24985 and 0.25004 as compared 

with the value #*=0.25004 given on page E21.

For many geological problems the age, £=1,385 m.y., 

would be acceptable. This age could then be reported 

for the assumption that the trial age discordancies 

were a result of an incorrect common lead correction 

and a failure to include a correction for original radiogenic lead. If, however, an additional refinement in 

the age is required, new values must be chosen for 

fi and/a from table 2 where tt now lies between the age 

limits, 1,385 <^ < 1,480. Selecting from table 2 

values of/! and/2 for ^=1,410, the recalculated age 

using equation 40 is t= 1,408 m.y. As this recalculated 

age now lies close to but below £1= 1,410, the concordant age must fall between 1,408 and 1,385 m.y. 

The relatively close agreement between the recalculated 

t and the second choice for ti would suggest that for 

most geological problems an additional age calculation 

would be unnecessary. The final age would be rounded 

down from 1,408 m.y. to the nearest 5 m.y. and reported as £=1,405 m.y. As has been mentioned, the 

lead-uranium ratios used in the example on page E21 

were chosen for £=1,400 m.y.

For certain geologic problems however, it may be 

desirable or necessary to evaluate further the assumption that the discordant trial ages are a consequence of 

errors in the correction made for the lead originally 

present in a group of related-age samples. If sufficient 

isotopic data are available from the area, it may be 

possible to compare the measured Pb207/Tb206 ratio of 

the lead extracted from associated radiogenically enriched nonradioactive minerals with the calculated R*. 

In addition, the measured Pb208/Pb204 and Pb20VPb204 

(or Pb206/Pb208 and Pb207/Pb208) ratios of the lead extracted from the associated nonradioactive minerals 

and the calculated 'N?a , 'N?a ratios (eq. 30, 31) can 

be plotted on the same graph. If the calculated R* 

and measured Pb207/Pb206 ratios are in reasonable 

agreement, and if the calculated 'N^* and fN°* lie on 

the same line as the measured Pb206/Pb204 and Pb207/ 

Pb206 ratios, these facts may be used as additional
METHODS FOR EVALUATING DISCORDANT LEAD-ISOTOPE AGES E25

evidence for the evaluation of the discordant trial 

ages in terms of original radiogenic lead.

To calculate 'N^ and 'N%* it is first necessary to 

find the value of t such that t approximates very closely 

ti} or #*(a_ 6) =.K*(a_ c) . If computing facilities are not 

available, additional recalculations using equation 40 

must be made.

Accepting the exact solution of equation 39 or 40 

as £=1,400 m.y., equation 37 may be used to obtain R* 

and equation 30 to calculate 'N^*. For sample A, the value 

of 'Nc *a is 1.9418 . Equation 31 is used to calculate W**, 

and the value obtained is 0.81804 . Similarly, for samples B and C, We*, W7J, WC6*, and 'N?c are respectively, 2.381 B , 0.92807 , 1.2685 , and 0.64957 . These values 

may be used to calculate the isotopic composition of 

the lead originally present in the samples and may now 

be plotted on the same graph with the measured Pb206/ 

Pb204 and Pb207/Pb204 ratios of the lead extracted from 

the associated nonradioactive minerals.

If the Pb207/Pb206 ratio, Rc , of the common lead is 

known from independent sources or is assumed, the 

amounts of common lead in the radioactive sample can 

be calculated fron the relation,

Knowing W6ca and Re, it is now possible to obtain 'N7ca , 

as well as fN*a and W7*. Finally, using Pb208= 1.0000 

and the Pb204/Pb208 ratio from the original isotopic 

analyses, the isotopic composition of the common lead 

may be calculated.

Thus, from the chemical and isotopic data on three 

closely related but unaltered radioactive samples that 

yield discordant trial ages, it is possible to calculate a 

single concordant age, t, without knowledge of either 

the amounts or the ratio of the common and original 

radiogenic lead assumed to be present in the sample. 

In addition, the ratio of the original radiogenic lead 

required to produce such a concordant age may be 

calculated as well as the isotopic composition of the 

lead originally present in the samples. These results 

combined with calculated ages obtained using other 

assumptions, additional isotopic data, and the field 

mineralogic relations, can then be used in the final 

evaluation of the age data.

GRAPHIC SOLUTION

The graphical concordant ages obtained by using 

normalized difference plots for not less than three 

samples, C, D, and E, formed at the same time, are 

shown in figure 4. Assuming, in addition, that the 

samples contain different amounts of lead and uranium 

and were contaminated by a common lead having a 

uniform Pb207/Pb206 ratio, a line passing through the

coordinates of the points (C D] and (C E) cuts the 

concordant age curve at t and t^ The slope of the line 

passing through (C D) and (C E}, divided by 137.7, 

is equal to R*, the Pb207/Pb206 ratio of the radiogenic 

lead lost or added at the time, t. The concordant age, t, 

is the age that would be obtained after correcting for 

the addition of an older generation of radiogenic lead 

formed, perhaps, in the time interval t 1\. The concordant age, ti, would correspond to the correction of 

samples C, D, and E for loss or gain of lead or uranium 

at one time in the past, t. Thus it is possible to obtain 

graphically concordant ages for a suite of at least three 

cogenetic uranium-bearing minerals without knowledge 

of either the amount or isotopic composition of the 

contaminating common lead, and either the effects of a 

single period of alteration or the presence of a yet 

older generation of radiogenic lead.

GENERAL EQUATIONS

The preceding discussion has been specifically 

directed at the problem of evaluating discordant leaduranium ages. Several of the equations that have 

been developed can be generalized, however, to aid 

in the evaluation of other types of discordant age pairs 

such as Pb206/U238-Pb208/Th232 and Pb207/U235-Pb208 / 

Th232 . Under certain limited conditions there is the 

possibility that the general form of the equations can 

be applied to such discordant age pairs as Pb206/U238- 

Sr87/Rb87 , Pb208/Th232-Sr87/Rb87, and perhaps others. 

It is not necessary for these pairs of radioactive parents 

and their stable daughter products to have been 

derived from the same radioactive mineral, although it 

is necessary for the minerals and their geo chronologic 

elements to meet the general conditions enumerated 

for the lead-uranium systems considered above.

An examination of the equations derived for the 

lead-uranium system suggests that the most useful 

general equations are those concerned with the correction for contamination by a common stable isotope 

which is indistinguishable from the stable daughter 

product produced by radioactive decay deposited at 

the time of mineral formation. The assumption for 

either the Pb-U235, Th232 or Pb-U238, Th232 systems 

that the same ratio of common daughters, Rac =I$), 

is present originally in two or more cogenetic mineral 

samples is geologically acceptable. The assumption, 

however, for other discordant age pairs that two or 

more cogenetic minerals would necessarily be originally 

contaminated by two radiogenically enriched components whose ratios were the same, that is, R*a=Rl, 

does not appear to be geologically justified. In the 

Pb-U system the ratio of the parents of Pb206 and 

Pb207, U238 and U235, are known to be essentially constant. The obvious possibility of local variations in
E26 SHORTEB CONTRIBUTIONS TO GENERAL GEOLOGY

the ratios of the concentrations of other radioactive 

parents such as U and Th, or even U and Rb could 

easily result in small local variations in the ratios of 

the radiogenic daughter products available to the 

solutions from which the new minerals were formed.

Equations 11, 12, 16, and 17 can be easily converted 

to the general form by making, where appropriate, the 

following changes in notation: R5 R1} RS R2 , R5t 

Rlt, and Rst ^t

wheTe f'Ri=Ndi/Npi, the ratio of the total number of 

atoms of stable radiogenic daughter product 

DI after correction (if necessary) for contamination of common D\ originally present 

to the total number of atoms of radioactive 

parent, PI, now present in the mineral 

sample, AI;

R2=Nd2 /Np2 , the ratio of the total number of 

atoms of a different stable radiogenic 

daughter product, D2 , after correction (if 

necessary) for contamination of common 

D2 C originally present to the total number of 

atoms of radioactive parent, P2 , now present 

in the same mineral sample AI or in a 

cogenetic mineral sample A2 ;

Rit= tNd1 /Npi, the ratio of the total number of 

atoms of stable daughter product, DI, to 

the total number of atoms of radioactive 

parent, Pit now present in the mineral 

sample A\;

R2t='Nd2 /Np2 , the ratio of the total number of 

atoms of stable daughter product, D2 , to the 

total number of atoms of radioactive 

parent, P2 , now present in the same mineral 

sample AI or in a cogenetic mineral sample; 

and where R, R*, Re, X1? \2, and t are defined as

R=Npi/Np2 , the ratio of the total number of 

atoms of the chemically different radioactive 

parents, PI and P2 , now present in the same 

mineral, Ai} or in a cogenetic pair of minerals, 

AI and A2 ;

R*=N*dJN*d2 , the ratio of the number of atoms 

of original radiogenic daughter product, 

D*, present at the time of mineral formation 

in mineral AI to the number of atoms of 

original radiogenic daughter product, Z?f, 

present at the time of mineral formation in 

the same mineral AI or a cogenetic mineral, 

A2 ;

Rc=Ncd1 /Ncd2) the ratio of the number of atoms of 

contaminating common daughter product, 

DI, present at the time of its formation in 

mineral Al to the number of atoms of contaminating daughter product, DC2 , present

at the time of mineral formation in mineral

AI or a cogenetic mineral A2 ', 

Xt=the decay constant of P-tXlO"10 y~l ; 

X2=the decay constant of P2 X 10~10 y~1 ', 

Xi5*X2 ; 

t=ihe age of mineral A and (or) cogenetic

mineral ^XIO10 years (tA=tAJ.

The general form of equation 26 cannot be as easily 

derived because, as has been mentioned, the term R is 

no longer the same constant for two or more samples, 

although it is known for each sample. Equation 26 

takes the form

(»- 1) !(»- 1) !""»! (»-2) I

/(2 » 1

-2)

A -RB) (XtX2)

(»_/\ (*L-i\ 

X> VXa ^ >

(w-3)\ i X(w-2)

+ (X!/2!+X2/2!) (RA -RB)

+ (RBRla 

where RA=NpiJNp2a} the ratio of the total number of 

the chemically different radioactive parents, 

Pia and P2a, now present in the same 

mineral A or in a cogenetic pair of minerals 

AI and A2 ;
METHODS FOR EVALUATING DISCORDANT LEAD-ISOTOPE AGES E27

the ratio of the total number of 

atoms of the chemically different radioactive 

parents, P1& and P26 now present in the same 

mineral B or in a cogenetic pair of minerals 

BI and B2 ;

Ria=Ndla/Npla, the ratio of the total number of 

atoms of stable daughter product, Dia, to 

the total number of atoms of radioactive 

parent, Piaj now present in mineral sample

-A,;

R2a=Nd2JNp2a , the ratio of the total number of 

atoms of stable daughter product, D2a, to 

the total number of atoms of radioactive 

parent, P2a, now present in mineral sample 

AI or in a cogenetic mineral sample A2 ;

Rii^Ndit/Npu,, the ratio of the total number of 

atoms of stable daughter product,Z?i & , to 

the total number of atoms of radioactive 

parent, PU, now present in mineral sample

X!

X2

t

the ratio of the total number of 

atoms of stable daughter product, D2tl , to 

the total number of atoms of radioactive 

parent, P26, now present in mineral sample 

BI or in a cogenetic mineral sample B2 ; 

decay constant of Pla and PuXlO"10 y~l ; 

decay constant of P2a and P2& X10~10 y~l ; 

the age of the mineral pair AI and BI or the 

pairs of cogenetic minerals AI, A2 , and Bi} 

B2 (tdj =*#!).

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o