Geochimica et Cosmochimica Acta, Vol. 62, No. 21/22, pp. 3437–3452, 1998
Copyright © 1998 Elsevier Science Ltd
Printed in the USA. All rights reserved
0016-7037/98 $19.00 1 .00
Pergamon
PII S0016-7037(98)00255-5
Uranium-thorium-protactinium dating systematics
H. CHENG,*,1 R. LAWRENCE EDWARDS,1 M. T. MURRELL,2 and T. M. BENJAMIN2
1
Minnesota Isotope Laboratory, Department of Geology and Geophysics, University of Minnesota, Minneapolis, Minnesota 55455, USA
2
Chemical Science and Technology Division, J514, Los Alamos National Laboratory, Los Alamos, New Mexico 87545, USA
(Received October 21, 1997; accepted in revised form July 24, 1998)
Abstract—With precise 234U, 230Th, and 231Pa data available, 230Th and 231Pa ages can now be tested
rigorously for concordancy. If the material is not concordant, the isotopic characteristics of this material may
be examined in some detail. Here, models similar to those used to describe the U-Pb system are evaluated for
use in U-Th-Pa studies, for the case in which initial 230Th and 231Pa concentrations are effectively zero. The
systematics of concordia plots in relation to models of variation in d234U, episodic U loss or gain, continuous
U loss or gain, and continuous 234U, 230Th and 231Pa gain or loss are considered for the case in which initial
U concentration is significant (for example, in many carbonate deposits). We also examine linear U uptake
models for the case in which initial U concentration is effectively zero (for example, in teeth and bones). Such
models should prove useful in interpreting data from materials that have behaved as open-systems. In
particular, these models may help constrain the nature of diagenetic processes, and in some situations it may
be possible to determine or constrain true ages with materials that have behaved as open-systems. Copyright
© 1998 Elsevier Science Ltd
ionization mass spectrometric (TIMS) methods for the measurement of 234U (Chen et al., 1986; 234U half-life 5 244,500 y,
Lounsbury and Durham, 1971, DeBievre et al., 1971) and
230
Th (Edwards et al., 1987a,b; Edwards, 1988) were developed, leading to significant increases in analytical precision in
230
Th dating. The use of TIMS for 230Th dating has resulted in
the establishment of precise chronologies for sea level changes,
continental climate shifts, and an absolute 14C calibration (e.g.,
Edwards et al., 1987a,b, 1993; Bard et al., 1990a,b; Ludwig et
al., 1992; Stein et al., 1993; Gallup et al., 1994). In addition,
precise initial d234U values (([234U/238U] 2 1) 3 1000, where
brackets indicate activity ratios) of marine samples have been
used to test for diagenetic alteration based on the assumption
that the d234U value of seawater is constant with location and
through time. However, 230Th studies still suffer from uncertainty about possible open-system behavior. For example,
many old coral samples have initial d234U values higher than
the modern marine value. Thus, it would be desirable to have
another chronometer available to test for age concordancy. The
231
Pa method, as first suggested by Sackett (1958), is a likely
choice, however, this method does not yield the required precision with decay-counting techniques. We have, therefore,
begun using TIMS techniques for 231Pa measurements in carbonates (Cheng et al., 1996; Edwards et al., 1997) using a
procedure originally developed at Los Alamos (Pickett et al.,
1994). This method provides data with precisions comparable
to the 230Th TIMS method, which makes it possible to use the
U-Th-Pa system to test for age concordancy and diagenetic
processes.
Up until now, there has been little motivation to establish the
basic systematics for U-Th-Pa dating with the same rigor as
seen for the U/Pb system, although these dating systems were
both developed during the 1950s to 1960s and share many
similarities. A number of previous studies have discussed UTh-Pa concordancy and applications to alpha-counting data
from coral, phosphorite, and U ore samples (Allégre, 1964; Ku,
1968; Szabo and Vedder, 1971; Ku et al., 1974; Szabo, 1979;
1. INTRODUCTION
230
Th is the second longest-lived intermediate daughter nuclide
(half-life t1/2 5 75383 years, Meadows et al., 1980) in the 238U
decay series, and 231Pa is the longest-lived intermediate daughter nuclide (t1/2 5 32760 y, Robert et al., 1969) in the 235U
decay series. Because U, Th, and Pa have different valences
(typically 41 and 61 for U, 41 for Th, and 51 for Pa) and
chemical affinities, significant fractionation can occur between
these elements during various geological processes. For example, Th and Pa have very low solubilities compared to U in
aqueous systems and are rapidly removed from fluids due to
their tendency to hydrolyze and sorb onto sinking particulates
(Gascoyne, 1992). Thus, minerals precipitated from waters can
have very low initial 230Th and 231Pa contents compared to
their parent U. The following return to equilibrium can then be
used as a measure of time. As the two daughters have different
characteristic timescales of ingrowth, they can be used as
independent measures of time, over timescales as long as 550
ka (230Th) and 200 ka (231Pa).
U-Th-Pa systematics have been used to obtain mineral formation ages since the 1950s. Barnes et al. (1956) measured
coralline 230Th and 238U by alpha-counting. Sackett (1958) first
suggested the use of both 231Pa and 230Th methods to date
carbonate samples. During the 1960s, the determinations of
230
Th, 231Pa, 234U, and 238U using alpha-counting methods
were applied to numerous dating applications (Allégre, 1964;
Thurber et al., 1965; Kaufman and Broecker, 1965; Broecker
and Thurber, 1965; Sakanoue et al., 1967; Rosholt, 1967; Ku,
1968; Szabo and Rosholt, 1969). In the 1970s and early 1980s,
the application of such techniques slowed because the precision
of the alpha-counting methods was insufficient to resolve many
dating problems, including the evaluation of the diagenetic
alteration of carbonate samples. In the late 1980s, thermal
*Author to whom correspondence
([email protected]).
should
be
addressed
3437
3438
H. Cheng et al.
Roe et al., 1982; Veeh, 1982; Birch et al., 1983; Burnett and
Kim, 1986; O’Brien et al., 1986; Veeh and France, 1988;
Kaufman and Ku, 1989; Kaufman et al., 1995). In general, the
differing effects of diagenesis were not considered in the detail
that is now necessary, because, in most cases, the available
precision was insufficient to resolve these differences. Before
this work, Allegré (1964) studied the equations for episodic U
gain for the case where initial [234U/238U] was unity and
Kaufmann and Ku (1989) and Kaufmann et al. (1995) studied
the case of continuous U gain and loss, also for the case in
which initial [234U/238U] was unity. In both cases these models
were used to interpret data from U-rich minerals. In a third
study, Rosholt (1967) found, and later confirmed (Szabo and
Rosholt, 1969), excess 231Pa relative to 235U in a molluscan
shell. They proposed a open-system model with two reservoirs
of U to explain this 231Pa excess (see Kaufman et al. (1971) for
a critical evaluation of this model).
Possible open-system behavior of U and Th isotopes has
been evaluated with TIMS 230Th data (Edwards et al., 1987a,b;
Edwards, 1988; Chen et al., 1991; Bard et al., 1991; Hamelin et
al., 1991; Henderson et al., 1993; Gallup et al., 1994). With
TIMS 231Pa data now available (Pickett et al., 1994; Cheng et
al., 1996; Goldstein et al., 1996; Holden et al., 1996; Pickett
and Murrell, 1997; Edwards et al., 1997), it is critical to
examine U-Th-Pa systematics in more detail. In this paper, we
present several basic models for the U-Th-Pa system using
[231Pa/235U] vs. [230Th/238U] or [231Pa/235U] vs. [230Th/234U]
diagrams which are analogous to 206Pb/238U vs. 207Pb/235U
concordia diagrams. This framework can potentially improve
our understanding of U-series dating results and lead to better
constraints for Quaternary chronology. It may also help us
understand the nature of diagenetic processes in various settings.
activity is defined as ai 5 liNi, where li and Ni are the decay
constant and the number of atoms for nuclide i, respectively.
2.2. Concordia Diagrams
Given the above conditions, the basic age equations for a
closed U-Th-Pa system are as follows (modified from Bateman,
1910; see also Ivanovich and Harmon, 1992 for a general
discussion of these equations):
F G
231
Pa
5 1 2 e2l231xt (231Pa age equation)
U
(1)
235
F G
234
U
5 11
U
238
SF G D
234
238
U
U
2 1
0
3 e 2l2343t ~ 234U/238U age equation)
F G
230
Th
5 12 e2l2303t 1
U
238
SF G D S
234
238
U
U
21 3
D
l 230
l 230 2 l 234
3 ~1 2 e2(l230-l234)3 t)(230Th/238U age equation)
F G F G S
S D
230
Th
5
U
234
238
234
(2)
(3)
S F GD
U
3 1 2 e2l2303t)1 1 2
U
238
234
U
U
l230
3 ~1 2 e2(l2302l234)3t!
l2302l234
3
~ 230Th/234U age equation!
(4)
2.1. Assumptions and Conventions
YF
F G F G
SF G D S
G
In the following models, a number of initial conditions and
conventions are adopted. (1) Initial 230Th and 231Pa are assumed to be zero. This is an important assumption that simplifies our theoretical treatment but also limits the use of our
models to materials for which this assumption is valid. Studies
of initial 231Pa levels, how one might correct for initial 231Pa,
relationships among initial 231Pa, initial 230Th, and 232Th concentrations, as well as the theoretical basis for interpreting
U/Th/Pa isotopic data for materials with significant initial
230
Th and 231Pa represent a class of problems which remain to
be studied and are beyond the scope of this contribution.
Although many materials may contain significant initial 230Th
or 231Pa (Kaufman and Broecker 1965; Ku et al., 1979;
Schwarcz, 1980), many calcite, aragonite, phosphorite, and
tooth or bone samples have been shown to contain trivial initial
230
Th or 231Pa (e.g., Edwards, 1988; Edwards et al., 1988;
Cheng et al., 1996; Goldstein et al., 1996; Edwards et al.,
1997). Our analysis aims to aid in the interpretation of data
from the latter types of materials. (2) 231Th, 234Th and 234Pa are
assumed to be in secular equilibrium at the time of sample
formation. (3)li 2 l238 5 li; li 2 l235 5 li; e2l238t 5
e2l235t 5 1, where li is the decay constant for nuclide i and t
is time. (4) Ratios in square brackets are activity ratios where
Where the subscript 0 denotes initial values.
From these equations, it is apparent that the U-Th-Pa system
is similar in many ways to the U-Pb system. In both cases, the
parents of the two decay schemes are U. Although the two
daughters, 231Pa and 230Th, are not the same element as in the
U-Pb system (daughters both Pb), they have similar chemical
properties in aqueous systems. It follows that in addition to
231
Pa and 230Th ages, it is also possible to determine 231Pa/
230
Th ages (Eqn. 5) with systematics analogous to 207Pb/206Pb
ages for the U/Pb system.
A significant difference between the U-Th-Pa and the U-Pb
systems is that the temporal change in the 230Th/238U and
230
Th/234U values depends on the initial d234U values of the
sample. Thus, different concordia curves exist for different
initial d234U values. Figure 1 gives [231Pa/235U] vs. [230Th/
234
U] and [231Pa/235U] vs. [230Th/238U] concordia diagrams for
different initial d234U values. In the [231Pa/235U] vs. [230Th/
238
U] diagram, the differences in the concordia curves are more
significant for different d234U values. This is because 238U is
essentially constant over these timescales and higher 234U/238U
ratios result in a relatively greater change in [230Th/238U] with
time (Fig. 1a). In contrast, the [231Pa/235U] vs. [230Th/234U]
diagram displays smaller differences between curves. This is
2. MODELS
231
235
230
238
Pa
5
Th
U
3 ~1 2 e2l2313t!
U
234
1
238
~1 2 e2l2303t!
D
U
l 230
21 3
U
l 230 2 l 234
3 ~1 2 e2(l2302l234)3t! (231Pa/230Th age equation)
(5)
U-Th-Pa dating
3439
Fig. 1. [230Th/238U] vs. [231Pa/235U] (a) and [230Th/234U] vs. [231Pa/235U] (b) concordia diagrams using half-lives of
75,383 years and 32,760 years for 230Th and 231Pa, respectively. The curves describe the evolution of the U-230Th-231Pa
system with initial activity ratios of [230Th/238U] (or [230Th/234U]) 5 0 and [231Pa/235U] 5 0, contoured with different initial
d234U values ranging from 0 to 1500. The horizontal lines are isochrons and the associated numbers indicate the ages in
thousands of years (ka). Diagram (a) is much more sensitive to initial d234U values than diagram (b). For evaluating
variations in d234U, diagram (a) may be more useful. For minimizing the effect of variations in d234U on age determinations,
diagram (b) may prove more useful.
because the more similar half-lives for 234U and 230Th result in
smaller changes in the 230Th/234U ratios (Fig. 1b). Both diagrams have distinct advantages. For example, if the effects on
d234U due to diagenesis are of interest, Fig. 1a will maximize
the sensitivity to variations in d234U. Alternatively, in order to
minimize the effect of initial d234U values on age determinations, Fig. 1b may prove useful. It will allow for age estimates
of marine samples using a single concordia curve, because the
initial d234U value for marine samples should be close to 150,
and the possible variations are indistinguishable in this diagram.
Uncertainties in the decay constants of 230Th and 231Pa lead
to uncertainties in the location of concordia. Direct measurements of the decay constants currently yield the following 2s
fractional errors: for 230Th, 68‰ (Meadows et al., 1980), and
for 231Pa, 67‰ (Robert et al., 1969). Recent measurements on
materials assumed to be in secular equilibrium (Cheng et al.,
1997) give a 230Th decay constant within error of that reported
by Meadows et al. (1980), but constrain the error to 63‰. The
propagation of errors in decay constants depends on the types
of standards used for each nuclide. For instance, if gravimetric
U and Th standards are used, the fractional error in 230Th age
due to error in the 230Th decay constant is less than the
fractional error in the decay constant for ages significantly less
than 105 y, but greater than the fractional error in the decay
constant for ages significantly larger than 105 y (Edwards et al.,
1987). However, if materials assumed to be in secular equilibrium are used for U and Th standards, the fractional error in age
due to error in the 230Th decay constant is similar to the
fractional error in the decay constant regardless of age (Ludwig
et al., 1992). For U and Th, both gravimetric and secular
equilibrium standards are used by the scientific community
today. However, for Pa only secular equilibrium standards are
used as, to the best of our knowledge, gravimetric Pa standards
do not exist. For this type of standardization, the fractional
error in 231Pa age due to error in the 231Pa decay constant is
exactly equal to the fractional error in the decay constant
regardless of age.
In Fig. 2, the thickness of the curve reflects the maximum
uncertainty in the position of the concordia curves due to the
uncertainties in the decay constants of 230Th and 231Pa, given
that analytical measurements of both abscissa and ordinate are
based on secular equilibrium standards. In the y-direction
([231Pa/235U]), the error in the position of concordia varies
from 67‰ (the fractional error in the 231Pa decay constant) at
zero age to less than 62‰ at .100 ka. In the x-direction
([230Th/238U] or [230Th/234U]) the error in the position of
concordia ranges from 63‰ (the fractional error in the 230Th
decay constant) at zero age to less than 61‰ at .250 ka. In
both abscissa and ordinate, the error in the position of concordia approaches zero as age approaches infinity and as the
activity ratios approach unity.
Figure 3 is analogous to Fig. 2, except that the error in the
x-direction is calculated assuming standardization to gravimetric Th and U standards. As the standardization is essentially to
a known atomic ratio of Th to U (as opposed to a known
activity ratio of Th to U as illustrated in Fig. 2), we use atomic
ratios on the abscissa of Fig. 3. In these diagrams, the uncertainties in the position of concordia in the ordinate ([231Pa/
235
U]) are as same as in Fig. 2. In the abscissa, uncertainties in
the position of concordia, increase from less than 1‰ at ages
,60 ka to 3‰ at ages . 500 ka. The error in the location of
concordia due to the uncertainty in the 230Th decay constant
vanishes as age approaches zero. As age approaches infinity,
3440
H. Cheng et al.
Fig. 2. [230Th/238U] vs. [231Pa/235U] (a) and [230Th/234U] vs. [231Pa/235U] (b) concordia diagrams similar to Figs. 1a and
1b, showing the effect of the half-life uncertainties for 230Th and 231Pa. The initial d234U is assumed to be 150. The thickness
in the curve represents the maximum variation in the concordia curve positions due to the half-life uncertainties. The white
circles around 50 ka in (a) and (b) represent the uncertainty at this age caused by the uncertainties in the half-lives of 230Th
(75383 years with 2s error of 63‰) and 231Pa (32,760 years with 2s error of 67 ‰). 2301, 2311, and 2302, 2312
indicate 230Th and 231Pa half-lives 3‰ and 7‰ higher than the nominal values, and 230Th and 231Pa half-lives 3‰ and 7‰
lower than the nominal values, respectively. For example, using 2301 and 2312, the concordia curve will be shifted to the
upper left; using 2302 and 2311, the curve will be shifted to the lower right. When 2301 and 2311 or 2302 and 2312
are used, the position of concordia curve is close to the position of a curve using the nominal 230Th half-life of 75383 y and
231
Pa half-life of 32760 y; however, ages along the curve will be shifted to the lower left or the upper right, respectively.
The arrows indicate the direction of these variations. The uncertainty of the concordia positions in ordinate ([231Pa/235U])
decreases from 7‰ (the fractional error in the 231Pa decay constant) at zero age to less than 2‰ for ages .100 ka, and in
abscissa ([230Th/238U] or [230Th/234U]), from 3‰ (the fractional error in the 230Th decay constant) at zero age to less than
1‰ for ages .250 ka.
the fractional error in the location of concordia in the x-direction approaches the fractional error in the 230Th decay constant.
As the error in the position of concordia as well as the error in
assigning times to points on concordia (see captions for Figs. 2
and 3) are in some cases comparable to analytical errors, it may
be necessary to consider these sources of uncertainty in applications using concordia plots.
Theoretically, all measured 231Pa and 230Th ages represent
apparent mean ages of samples which have been deposited over
some period of time. These apparent ages are a little younger
than the true mean ages. This phenomenon occurs because the
rates of 231Pa and 230Th accumulation slow with time. Thus,
ages calculated from average 231Pa/235U and 230Th /238U (or
230
Th /234U) ratios do not correspond exactly to true mean ages.
Furthermore, the offset between the 231Pa and true mean age
and that between the 230Th and true mean age are in general
different. Measured mean 231Pa ages are younger than corresponding 230Th ages. The 231Pa and 230Th mean age differences are generally much smaller than the analytical errors.
However, if the sample has been deposited over a large (.50
ka) time interval (for example, if the sample growth rate is
extremely low), the difference in mean ages may need to be
considered for concordia problems.
d234U vs. [231Pa/235U] and d234U vs. [230Th/238U] (or
230
[ Th/234U]) are other examples of concordia diagrams for the
U-Th-Pa system (Fig. 4a,b). The d234U vs. [230Th/238U] dia-
gram has been widely used to evaluate possible diagenesis for
coral samples (i.e., Edwards, 1988; Ku et al., 1990; Bard et al.,
1991; Chen et al., 1991; Gallup et al., 1994). With the availability of TIMS data for 231Pa, the d234U vs. [231Pa/235U] plot
can now also be used to check for diagenesis.
Although the 231Pa dating method has a shorter time range
(;200 ka) than that of the 230Th dating method (;550 ka), it
still can be used to test for concordia and evaluate diagenesis
processes for samples older than 200 ka. For samples older than
about 200 ka, the [231Pa /235U] ratio must equal one, within
present analytical errors, if the sample has behaved as a closedsystem. Values other than one suggest gain or loss of either U
or 231Pa (see following discussion).
2.3. Episodic Uranium Loss or Gain Relative to
and 231Pa
230
Th
Allégre (1964) first provided an episodic model for the
U-Th-Pa system for the case in which [234U/238U] equals one.
This is similar to the episodic model for the U-Pb systematics
introduced by Wetherill (1965a,b). The basic assumption is that
the sample experienced open-system conditions for a very short
period (compared to true age of the sample) which resulted in
Pb loss or U gain. Minerals of the same age but experiencing
different degrees of Pb loss or U gain will fall on a straight line
with the upper intercept with the concordia curve correspond-
U-Th-Pa dating
3441
Fig. 3. 230Th/238U vs. [231Pa/235U] (a) and 230Th/234U vs. [231Pa/235U] (b) concordia diagrams. All notations and legends
are as same as those in Fig. 2, except the abscissas, 230Th/238U and 230Th/234U, which are atomic ratios instead of activity
ratios. As opposed to Fig. 2, which illustrates how errors in decay constants propagate if secular equilibrium standards are
used for both Pa/U and Th/U, this figure illustrates how errors in decay constants propagate if secular equilibrium standards
are used for Pa/U, but gravimetric standards are used for Th/U. The uncertainty of the concordia positions in ordinate,
([231Pa/235U]) is as same as Fig. 2. However, in this case, the error in the position of concordia in the abscissa (230Th/238U
or 230Th/234U), increases from less than 1‰ for ages ,60 ka to up to 3‰ (the fractional error in the 230Th decay constant)
for ages . 500 ka.
ing to the time of crystallization and the lower intercept corresponding to the time of the secondary open-system event.
The U-Th-Pa system has very similar features. The following
discussion refers to episodic U loss or gain, for the case where
U isotopes are not fractionated during loss and where any
gained U has the same isotopic composition as the sample. In
the discussion, we refer to U loss or gain, but these processes
are mathematically equivalent (and equally applicable) to Pa
Fig. 4. Plots of d234U vs. [231Pa/235U] (a) and d234U vs. [230Th/238U] (b), contoured in 231Pa age and 230Th age (vertical
and subvertical lines in (a) and (b), respectively) and initial d234U (subhorizontal curves in (a) and (b)). Diagram (b) has
been used previously to evaluate diagenesis. With 231Pa TIMS data now available, a similar plot for [231Pa/235U] may be
useful (a).
3442
H. Cheng et al.
and Th gain or loss, subject to the condition that Pa and Th are
gained in the same proportion as they are present in the sample
or that Pa and Th are lost without fractionation. If U loss or gain
occurs over a very short period (tD years ago, D for diagenesis),
samples of the same primary age (tP years, P for primary) with
varying degrees of U loss or gain will also fall on a straight line
in a [231Pa/235U] vs. [230Th/238U] or [231Pa/235U] vs. [230Th/
234
U] concordia plot. The upper intercept with the concordia
curve corresponds to their true primary age (tP). A major
difference between this system and the U-Pb system is that the
lower intercept does not theoretically correspond to the exact
age of the secondary event (tD) unless initial d234U is zero (i.e.,
[234U/238U] 5 1). The basic equations for the episodic model
for U-Th-Pa evolution are:
F G
231
Pa
5 ~1 2 e2l2313(tP2tD)) 3 e2l2313tD
U
235
3 F 1 ~1 2 e2l2313tD!
F G F
230
Th
5 ~1 2 e2l2303(tP2tD)! 1
U
238
3 e2l2343(tP2tD) 3
S
SF G D
234
238
D
U
U
21
0
l 230
3 ~1 2 e2(l2302l234)3(tP2tD)!
l 230 2 l 234
SF G D
234
3 e2l2303tD 3 F 1 ~1 2 e2l2303tD! 1
234
238
230
Th
5
U
234
234
230
Th
U
238
Th
U
238
234
238
238
U
U
G
21
0
l230
3 ~1 2 e2(l2302l234)3tD!
l2302l234
U
511
U
230
5
S
U
U
238
D
F G SF G D
F G F GYF G
F G Y F SF G D
3 e2l2343tP 3
(6)
2 1 3 e2l2343tP
(7)
(8)
0
U
U
234
11
238
U
U
2 1 3 e2l2343tP
0
G
(9)
Where at time tD, U loss or gain follows: Uold/Unew 5 F. Where
Uold and Unew are U atoms per unit sample before and after tD,
respectively. tP is the sample’s primary age. F is related to the
fraction of U lost (.1) or gained (,1) during the diagenetic
event at time, tD.
Using these equations, it can be shown that the derivatives
d[231Pa/235U]/d[230Th/238U],
d[231Pa/235U]/d[230Th/234U],
234
238
231
235
d[ U/ U]/[ Pa/ U] and d[234U/238U]/[230Th/238U] are
all independent of F. Therefore, samples of the same age with
different degrees of U loss or gain occurring at tD will plot
along straight lines in diagrams of [231Pa/235U] vs. [230Th/
238
U] (Fig. 5a), [231Pa/235U] vs. [230Th/234U] (Fig. 5b), d234U
vs. [231Pa/235U] (Fig. 6a), and d234U vs. [230Th/238U] (Fig. 6b).
The upper intersection of the line with the concordia curve
corresponds to F 5 1 (no U loss or gain) and indicates the
sample’s primary age (tP). If d234U 5 0, the lower intersection
of the line with the concordia curve corresponds to F 5 0 (all
230
Th and 231Pa lost or infinite addition of U) and indicates the
time of open-system behavior (tD). When d234U is negative, the
point corresponding to tD and F 5 0 will actually lie to the
upper right of the lower intercept. When d234U is positive, this
point will lie to lower left of the lower intercept. The larger the
initial d234U offset from 0, the larger the distance between the
point corresponding to tD and F 5 0 and the lower intercept.
However, in many cases the age of most interest is the true age
(tP, the upper intercept point). Furthermore, if initial d234U is
close to 0, for example between 0 and 200 (the case for marine
samples), the lower intercept will be indistinguishable from the
F 5 0 point in a [231Pa/235U] vs. [230Th/234U] diagram (Fig.
5b). Other graphical methods common to the episodic model
for the U-Pb system can also be applied to the U-Th-Pa system.
For example, the distance of the data points from the upper
intercept point represents the fractional U loss or gain relative
to 230Th and 231Pa.
For the case of relative U loss, the different ages have the
following relationship: 231Pa age . 230Th age . 231Pa/230Th
age . true age. For the case of relative U gain, 231Pa age
, 230Th age , 231Pa/230Th age , true age. Thus, in these types
of open systems, if analytical imprecision is negligible, the
231
Pa/230Th age is closest to the true age. In practice, constraints from 231Pa/230Th ages are most useful for samples
older than about 10 ka years as 231Pa/230Th ages are significantly less precise than either 231Pa or 230Th ages for samples
younger than 10 ka (because the fractional change in 231Pa/
230
Th ratio with time is very small in this interval).
We can estimate a sample’s 230Th, 231Pa, and 231Pa/230Th
ages through graphical methods using concordia plots (Fig. 1).
Note that if a sample has been altered these ages will not in
general be identical to the true age of the sample. In either a
[231Pa/235U] vs. [230Th/238U] plot (Fig. 1a) or a [231Pa/235U]
vs. [230Th/234U] plot (Fig. 1b), the 231Pa age is given by the
intersection between concordia and a horizontal line through a
sample point. For estimating 230Th and 230Th/231Pa ages, the
initial d234U must be known, either independently or by using
the estimated age from a concordia plot to calculate an initial
d234U from a measured value. One can then select the concordia curve that corresponds to the initial d234U value. Given this
curve in either a [231Pa/235U] vs. [230Th/238U] plot (Fig. 1a) or
a [231Pa/235U] vs. [230Th/234U] plot (Fig. 1b), the 230Th age is
given by the intersection between concordia and a vertical line
through a sample point and the 231Pa/230Th age is given by the
intersection between concordia and a line between a sample
point and the origin. [231Pa/235U] vs. [230Th/234U] plots are
generally the most useful for graphically estimating 230Th and
231
Pa/230Th ages as the position of concordia is not strongly
dependent on initial d234U value in this type of plot.
2.4. Continuous Uranium Loss or Gain Relative to
and 231Pa
230
Th
If U loss or gain occurs as a continuous process with no
change in the U isotopic composition, the behavior of the
U-Th-Pa system is graphically similar to the diffusion model
for U-Pb systematics (Nicolaysen, 1957; Tilton, 1960; Wasserburg, 1963). Kaufmann and Ku (1989) and Kaufmann et al.
(1995) examined this process for the case that the initial d234U
value and the d234U value of added U were both zero. We
examine this process with initial d234U as a variable, but
stipulate that gained or lost U may not change the U isotopic
U-Th-Pa dating
3443
Fig. 5. [230Th/238U] vs. [231Pa/235U] (a) and [230Th/234U] vs. [231Pa/235U] (b) concordia diagrams showing the effect of
episodic U loss or gain relative to 230Th and 231Pa 20 ka ago (tD) for a suite samples with the same crystallization age of
100 ka (tP). Upon initial precipitation 100 ka ago, the whole suite of samples lie on the origin. Over the next 80 ka, the
isotopic composition of the suite is assumed to evolve through closed-system radioactive decay and ingrowth, following
concordia to the 80 ka point. At that time (20 ka ago, tD), the suite of samples undergoes episodic diagenesis. Some samples
gain U (F , 1) and some lose U (F . 1). The squares represent the isotopic composition of the suite of samples immediately
after the open-system episode 20 ka ago (tD). The square on concordia at 80 ka represents a sample in the suite that has not
gained or lost any U. The squares above concordia show the isotopic compositions samples that have lost U (20% for the
first square above concordia (F 5 1.2), 40% for the second square above concordia (F 5 1.4). The squares below concordia
show the isotopic composition of samples that have gained U. The first square below concordia shows the isotopic
composition of a sample, which after the episode has 80% primary U and 20% diagenetic U (F 5 0.8). The second square
below concordia shows the isotopic composition of a sample that has 60% primary U and 40% diagenetic U (F 5 0.6), and
so on. The square that plots on the origin has 0% primary U (infinite addition of diagenetic U, F 5 0). The dashed line in
each diagram connects the squares and is mathematically linear. After this episode of U loss and gain, the suite is assumed
to again behave as a closed-system. The square points evolve by radioactive decay and ingrowth along the dotted lines to
the corresponding circles over a period of 20 ka. At all times in the intervening 20 ka, the points lie on a line that intersects
concordia at a time that represents the primary age of the suite (upper intercept with concordia) and a time that is generally
close to the time of the diagenetic episode (if initial d234U is reasonably close to zero; see text, lower intercept with
concordia). The circles represent the isotopic composition of the suite of samples after the intervening 20 ka of
closed-system behavior. In each diagram, the fine solid line connects the open circles and intersects concordia at the time
of primary crystallization and close to the time of episodic diagenesis. In this model, samples that plot above concordia,
represent samples that lost U relative to both 230Th and 231Pa at tD, and samples that plot below concordia, gained U relative
to both 230Th and 231Pa at tD. The distance between a sample point and upper intercept point is proportional to the degree
of its U loss, for samples that plot above concordia, or to the fraction of diagenetic U contained in the sample, for samples
that plot below concordia.
composition of the material. In our model, we refer to U loss or
gain, however, these processes are mathematically equivalent
to gain of both 231Pa and 230Th or loss of both 231Pa and 230Th
if the gain or loss causes no shift in the 231Pa/230Th ratio in the
sample. The equations for our model are given in the next
section. Samples of the same age will lie along a curve,
instead of a straight line, in [231Pa/235U] vs. [230Th/238U]
and [231Pa/235U] vs. [230Th/234U] diagrams (Fig. 7a and 7b).
However, this curve is nearly linear near its upper intercept
with concordia. In plots of d234U vs. [231Pa/235U] and d234U
vs. [230Th/238U], samples of the same age will lie on lines
(Fig. 8a and 8b). In diagrams of [231Pa/235U] vs. [230Th/
238
U] and [231Pa/235U] vs. [230Th/234U], the lower intercept
now has no temporal significance; however, the upper intercept still corresponds to the primary age. Because the curve
is indistinguishable from a straight line near the upper intercept (Fig. 7a,b), it may be possible to estimate the primary
age by graphical methods. In this model, the 230Th, 231Pa
and 231Pa/230Th ages have the same relationship as in the
episodic model described above for both U loss and gain
cases.
2.5. Continuous Addition or Loss of
and 231Pa
234
U,
230
Th
Gallup et al. (1994) formulated a 230Th and 234U continuous
input model to explain U-series results obtained for Barbados
corals. We present here a general model for continuous 234U,
230
Th, and 231Pa input together with continuous loss or gain of
U. The basic differential equations for this model are:
d 238U
5 2l238 3 238U 1 Ru 3 238U
dt
(10)
d 235U
5 2l235 3 235U 1 Ru 3 235U
dt
(11)
3444
H. Cheng et al.
Fig. 6. Plots of d234U vs. [231Pa/235U] (a) and d234U vs. [230Th/238U] (b). (See caption for Fig. 4). The episodic
assumptions, symbols and lines are same as in Fig. 5. Each of the squares and open circles in this figure correspond to an
analogous square or circle in Fig. 5. The intercept of the horizontal fine solid line with the concordia curve (initial
d234U 5 150) gives the true crystallization age for a suite of samples of the same age. Points that lie to the right of the
concordia curve indicate U loss (F . 1) or gain of both 230Th and 231Pa. Points that lie to the left of concordia curve indicate
U gain (F , 1) or loss of both 230Th and 231Pa. No information about the time of U loss or gain can be obtained in diagrams
such as these. In addition, the initial d234U must be known in order to select the correct concordia curve to be used for a
suite samples.
d 231Pa
5 l235 3 235U 2 l 231 3 231Pa 1 R231
dt
d
(12)
234
Ut 5
234
U
5 l 238 3 238U 2 l 234 3 234U 1 Ru 3 234U 1 R234
dt
d 230Th
5 l 234 3 234U 2 l 230 3 230Th 1 R230
dt
2
(13)
Tho 5 Pao 5 0,
Tht 5
2
U0 5 137.88 3 235Uo,
238
F G
238
U
U
235
231
5 constant,
Ut 5 238Uo 3 e(Ru2l238)3t 5 238Uo 3 eRu3t
(15)
Ut 5 235Uo 3 e(Ru2l235)3t 5 235Uo 3 eRu3t
(16)
S
2l2313t
3e
D
R231
1
l231
0
l 238 3 238U
.
l234
U
U
3
0
l238 3 238Uo
l234
l238 3 238Uo
l 234
R234
3
2
l234 2 Ru
l234
Ru 2 l234 1 l230
l 238 3 238Uo
1
(Ru 1 l230
234
238
U
U
o
l 238 3 238Uo
l238 3 238Uo
R234
2
2
l234
l234 2 Ru
l234
3
l 234
l234 3 R234
R230
1
1
Ru 2 l234 1 l230 l230 l230 3 (l234 2 Ru)
3 e2l2303t 1
(17)
234
238
3
0
l235 3 235Uo
R231 l235 3 235Uo
Pat 5
3 eRu3t 2
1
Ru 1 l231
l231
Ru 1 l231
3
FF G
G
SF G
F
D
l238 3 238Uo
3 eRu3t 1
l234
3 e2(l2342Ru)3t 2
we obtain the following analytical solutions to Eqns. 10214:
238
U
238
U
(18)
(14)
Using the following initial and boundary conditions,
234
FF G
G
234
l238 3 Uo
R234
R234
3 e2(l2342Ru)3t 1
2
l234 2 Ru
l234
l234 2 Ru
238
234
230
l238 3 238Uo
3 eRu3t 1
l234
l234 3 R234
R230
1
l230 l230 3 (l234 2 Ru)
G
(19)
Where l230, l231, l234, l235, and l238 are the decay constants
(with units of inverse time) for 230Th, 231Pa, 234U, 235U, and
238
U. R230, R231, and R234 are input rates (with units of atoms/
U-Th-Pa dating
3445
Fig. 7. [230Th/238U] vs. [231Pa/235U] (a) and [230Th/234U] vs. [231Pa/235U] (b) concordia diagrams representing continuous
U loss or gain. Suites of 150 ka and 80 ka age samples with different U loss or gain rates plot along the fine solid and dashed
curves, respectively. The upper intercept of each curve with concordia represents the true age of the suite. Samples that
follow this model and plot on the portion of each dashed or fine solid line above concordia have undergone continuous U
loss relative to Pa and Th and samples that plot on the portion of each curve below concordia have undergone continuous
U gain relative to Pa and Th. The rate of U gain is greatest for samples that plot on the portion of each curve near the origin,
whereas the rate of U loss is greatest for samples that plot on the portion of each curve furthest from the origin. The diagram
is graphically similar to the diffusion model for the U-Pb system. It is important to note that the curve near the upper
intercept is close to a straight line, which may allow the use of graphical methods to obtain the true age of a set of samples
of the same age by extrapolation to its upper intercept with concordia. However, the extrapolation of the near linear portion
of each curve to its lower intercept with concordia will yield a meaningless age, as has been demonstrated for the U-Pb
system (Tilton, 1960).
time) of 230Th, 231Pa, 234U from outside of the system. Ru is
the fraction (per unit time) of U loss (negative) or gain (positive) relative to 230Th and 231Pa, and it is assumed to be much
greater than l235 and l238 (Ru .. l235 and Ru .. l238). t is
the age of sample. 230Th, 231Pa, 234U, 235U, and 238U represent
atoms of 230Th, 231Pa, 234U, 235U, and 238U. Subscripts t and o
indicate time t and t 5 0, respectively.
If R231 5 R230 5 R234 5 0 and Ru Þ 0, the equations
describe the continuous U loss (negative Ru) and gain (positive
Ru) model (see Figs. 7 and 8). If R230 ' R234, and Ru and
R231 5 0, the equations describe the 230Th and 234U continuous
input model of Gallup et al., 1994.
If R231, R230, and R234 Þ 0, and Ru 5 0, the equations
describe the 234U, 230Th and 231Pa continuous input model
shown in Figs. 9 and 10. Data for samples of the same age will
lie along a curve. The intercept between this curve and the
concordia curve is the primary age of the sample set. The
portion of curve near the upper intercept with concordia is very
close to a straight line with a slope dependent on the R231, R230,
and R234 values. In this continuous addition model, the 231Pa
age will be greater than the true age, the 230Th age and the
231
Pa/230Th age could be greater or less than true age, depending on the input ratios of R230/R234 and R231/R230. Here, the
231
Pa age provides the best upper limit on the true crystallization age.
This model can be extended to models for 231Pa, 230Th, and
234
U continuous losses (R231, R230, and R234 are negative) as
shown in Figs. 9 and 10 (the very short curves from concordia
curve down to lower left). For this case, the 231Pa age will be
less than the sample’s true age. As with the addition model, the
230
Th and the 231Pa/230Th ages also could be greater or less
than the true age, depending on the relative loss rates, i.e.,
R230/R234 and R231/R230.
2.6. Linear Uptake of Uranium, with No Initial Uranium
Ikeya (1982) suggested two U-accumulation models based
on electron spin resonance (ESR) analysis of teeth, which start
with essentially no U, but pick up U diagenetically. Such
models could be equally applicable to other materials such as
mollusks and bones, which also start with essentially no U. The
two models are (1) early uptake (EU) of U in which U accumulated within a short time span after it was buried, (2) linear
uptake (LU) of U in which U addition was continuous and
constant. He favored the LU model. Ku (1982) outlined the
230
Th age, 231Pa age and true age differences for the LU model
using a plot of the true age vs. the average apparent age.
Subsequently, the LU model has been discussed and applied to
many tooth and bone U-series and ESR dating studies (e.g.,
Grün et al., 1988; Chen et al., 1994; Grün and McDermott
1994; Swisher et al., 1996). These studies focused mainly on
ESR dating and corrections. Kaufmann et al. (1995) examined
the LU model with respect to U-Th-Pa systematics, under the
assumption that d234U is constant and equal to zero. Here, we
describe the basic behavior of the U-Th-Pa system in the LU
model, with the d234U of added U as a constant not equal to
3446
H. Cheng et al.
Fig. 8. Plots of d234U vs. [231Pa/235U] (a) and d234U vs. [230Th/238U] (b) (see caption for Fig. 4), showing continuous U
loss or gain relative to 230Th and 231Pa. Suites of 150 ka and 80 ka age samples with different U loss or gain rates lie along
the fine solid and dashed horizontal lines, respectively. Each of the dotted curves illustrates how the isotopic composition
of a sample with a specific U loss or gain rate changes with time.
zero. The d234U of the material is a variable, which is allowed
to change as a function of time following mass balance and
radioactive decay constraints. The basic differential equations
are
d
238
d
235
U
5a
dt
231
Pat 5
234
Ut 5
d
U
5b
dt
(21)
Pa
5 l2353235U 2 l 231 3 231Pa
dt
(22)
230
Th
5 l2343234U 2 l 230 3 230Th
dt
(24)
At t 5 0:
238
Tho 5
Pao 5
Uo 5
235
Uo 5
234
Uo 5 0
U 5 137.88 3 235U (or a 5 137.88 3 b)
S D
234
238
U
U
5 c
input
we obtain the following analytical solutions to Eqns. 20224:
Ut 5 a 3 t
(25)
Ut 5 b 3 t
(26)
238
235
S
S
D
l 238 3 a
a3c
)2
l234
l2342
(28)
D
l238 3 a
a3c
3 e2l2343t
2
l234 3 (l230 2 l234) (l230 2 l234)
1
F
1
l238 3 a
l238 3 a
a3c
2
3 e2l2303t 1
l230 3 l234
l230
l230
a3c
l238 3 a
l238 3 a
2
1
l230 2 l234 l234 3 (l230 2 l234)
l2302
3t1
Using the following initial and boundary conditions,
238
Tht 5
(23)
d 234U
5 l238 3 238U 2 l 234 3 234U 1 a 3 c
dt
231
D
(27)
l238 3 a a 3 c
l238 3 a
2
3 e2l2343t 1
l2342
l234
l234
3t1
231
230
S
(20)
230
d
b 3 l235
3 (e2l2313t 1 l231 3 t 2 1)
l2312
G
l238 3 a
l238 3 a
a3c
2
2
l230
l230 3 l234
l2302
S
D
(29)
Where a and b are constant rates of 238U and 235U accumulation; c is the 234U/238U atomic ratio of the input U; t is the true
depositional age of the sample.
If U uptake has been linear, the standard 231Pa, 230Th and
231
Pa/230Th ages, which are equivalent to EU ages, will all be
younger than the true age of a sample. In [231Pa/235U] vs.
[230Th/238U], [231Pa/235U] vs. [230Th/234U] plots, d234U vs.
[231Pa/235U], and d234U vs. [230Th/238U] plots (Figs. 11 and
12), the data points lie below the concordia curves. This tendency becomes large when true ages are greater than about 20
ka. This deviation from concordia can be used to check the
U-Th-Pa dating
Fig. 9. [230Th/238U] vs. [231Pa/235U] (a) and [230Th/234U] vs. [231Pa/235U] (b) concordia diagrams. When 230Th, 231Pa and
U are added to (or lost from) samples continuously, the data for samples of the same age will lie along a curve (the short
solid or dotted curves). The intercept of this curve with concordia will indicate the true crystallization age. These curves
are indistinguishable from straight lines near the intercepts, and their slopes depend on the relative addition rates of 230Th,
231
Pa and 234U (R230, R234, and R231). In these two diagrams, two examples are given: R230 5 R234 5 20 * R231 (short dotted
curves) and R230 5 R234 5 50 * R231(short solid curves). These ratios were chosen because 20 is close to the production
ratio of 230Th to 231Pa and would be the approximate isotope ratio of a young carbonate (,10 ka). 50 is close to the
230
Th/231Pa ratio of secular equilibrium materials. The upper end and lower end of each short line represent 200,000
atom/year/g gain and 100,000 atom/year/g loss of 230Th for a sample with a 3 ppm 238U concentration, a typical
concentration for a coral. For reference, a coral with this 238U concentration produces about 1.3 million atoms of
230
Th/year/g by radioactive decay. Graphical methods can be used to obtain the true age for a suite of the samples of the
same age.
234
Fig. 10. Continuous 230Th, 231Pa and 234U addition models for plots of d234U vs. [231Pa/235U] (a) and d234U vs.
[230Th/238U] (b). The symbols are same as in Fig. 8. The dashed curves (b) represent the condition of R230 5 R234. All of
the thin dashed, dotted, and solid curves are indistinguishable from straight lines. It is possible to get information for a suite
of samples of the same age using graphical methods; however, the initial d234U values must be known. In this model, the
curves are generally not subhorizontal, which is different from simple U loss and gain models.
3447
3448
H. Cheng et al.
Fig. 11. [230Th/238U] vs. [231Pa/235U] (a) and [230Th/234U] vs. [231Pa/235U] (b) concordia plots. Solid curves are concordia
curves (initial d234U 5 150 and 1500) as in Fig. 1. Dashed curves are concordia plots assuming linear U (LU) uptake (initial
d234U 5 150 and 1500). The ages on LU model curves correspond to the true ages of samples for which uptake of U has
been constant with time. 230Th and 231Pa ages calculated assuming linear U uptake are older than those calculated assuming
closed-system behavior for the same measured [230Th/238U], [230Th/234U], and [231Pa/235U] values. Diagrams such as these
can be used to test for LU model behavior (isotopic compositions should lie on the dashed linear uptake concordia curves).
applicability of the LU model. In addition, Eqns. 20, 23, and 24
can be used to solve for a, c, and t; and Eqns. 21 and 22 will
provide b and t. In this way, age and input rate can be checked
for concordancy.
If the sample follows the LU model, the different standard
ages of a sample have the following pattern: true age $ apparent 231Pa/230Th age . apparent 230Th age $ apparent 231Pa
age. When the true age is less than 20 ka, the true age is nearly
twice that of the apparent 231Pa or 230Th ages (Figs. 11 and 12)
and about 1.5 ; 1.6 times that of the apparent 231Pa/230Th age.
Fig. 12. Plots of d234U vs. [231Pa/235U] (a) and d234U vs. [230Th/238U] (b). The concordia curves of the LU model (initial
d234U 5 150) are shown as dashed curves. They are below close-system concordia curves with the same initial d234U value
(solid curves). The offset increases with the age.
U-Th-Pa dating
3449
Fig. 13. [230Th/238U] vs. [231Pa/235U] (a) and [230Th/234U] vs. [231Pa/235U] (b) concordia plots for last interglacial corals
from Barbados (Gallup et al., 1994; Edwards et al., 1997). The solid ellipses represent 2s errors and the labels are sample
numbers. The heavy lines are best-fit lines calculated using the least squares method (y 5 0.926x 1 0.211 and y 5 1.462x
2 0.087 in diagrams (a) and (b), respectively). r is the correlation coefficient. Other symbols are as same as in Fig. 9. The
continuous U loss or gain lines, similar to Fig. 9, are plotted for comparison. The best-fit lines intersect concordia at ages
of 131 ka in diagram (a) and 126 ka in diagram (b). Of the two ages, the 126 ka age is the best estimate of the true age for
two reasons: (1) the effect of uncertainty in initial d234U on the age of the intercept is insignificant in diagram (b) and (2)
the angle of intersection between the best-fit line and concordia is much steeper in diagram (b) than in diagram (a).
The differences between true and apparent ages increase with
age (Figs. 11 and 12).
3. AN EXAMPLE
Here we present an example of the use of these concordia
plots using some of the first high resolution U-Th-Pa data. The
example uses five analyses of four last interglacial samples
(two subsamples of FU-1, Figs. 13 and 14) from Barbados (see
Gallup et al., 1994 for information about samples and the
original U/Th data and Edwards et al., 1997 for more recent
U/Th data and the U/Pa data). These samples should have
similar ages (the last interglacial), if they have behaved as
closed-systems. However, some of them, have been altered
during diagenesis as shown by the discordant 231Pa and 230Th
ages (FU-3 and UWI-16). If the diagenesis process follows one
of the models described above (Figs. 5210), then these data
will lie on a line in each plot (i.e., in Figs. 5210). The least
squares method was used to obtain best fit lines. The correlation
coefficients are between 0.9865 and 0.9983 for the data in four
types of plots ([231Pa/235U] vs. [230Th/238U], [231Pa/235U] vs.
[230Th/234U], d234U vs. [230Th/238U], and d234U vs. [231Pa/
235
U]. In the first two plots, all the data lie within analytical
error of a best-fit line and the correlation coefficients (0.9983
and 0.9924, Fig. 13) are very high. For the two plots with d234U
as the ordinate the correlation coefficients are also very high
(0.9913 and 0.9865), but slightly lower than in the other two
plots. Also, in the latter two plots, the data are not co-linear
within analytical error. The data are consistent with a 230Th,
231
Pa and 234U continuous input model, or a model of continuous loss of U coupled with continuous increase in d234U. As
the U concentrations of the discordant samples are similar those
of primary corals (see Min et al., 1995), the addition model
would appear to be more appropriate. The parameters that best
fit the data in a continuous input model are a 230Th/234U input
ratio is close to 1 (see Gallup et al., 1994) and a 230Th/231Pa
input ratio of about 35. Young carbonate material (,10 ka) has
a 230Th/231Pa ratio of about 20 (the production ratio for material with d234U close to zero), and old material (secular equilibrium material) has a 230Th/231Pa ratio of about 50. Thus, the
calculated 230Th/231Pa ratio of the added material is intermediate between the two extremes, suggesting that net fractionation of 230Th from 231Pa during the dissolution of the ultimate
source material and incorporation of the 230Th and 231Pa in the
corals was small. If fractionation was negligible, the data are
consistent with a source material of about the same age as the
corals. Thus, a plausible source for the added Th and Pa is
dissolution of carbonates of the same age as the coral.
If the initial d234U is 150, the ages of intercepts between
best-fit lines and concordia curves are 131 ka, 126 ka, 116 ka,
and 121 ka for plots of Figs. 13a,b, and 14a,b, respectively.
Among these plots, the [231Pa/235U] vs. [230Th/234U] diagram
provides the best upper intercept age (126 ka), largely because
it is not affected by small variations in initial d234U value. For
example, the upper intercept age in this example only changes
0.1% when the initial d234U value changes by 65‰ around
150 (from 145 to 155). However, for the other three plots, the
upper intercept ages shift 1.322.8% for the same shift in initial
d234U value. Furthermore, for the plots with d234U on the
ordinate, the correlation coefficients, although high, are smaller
than for the plots with [231Pa/235U] on the ordinate. Considering the latter two plots, in the [231Pa/235U] vs. [230Th/238U]
diagram, the angle of intersection between concordia and a
best-fit line is much shallower than in the [231Pa/235U] vs.
[230Th/234U] diagram. Thus, errors in the location of the best-fit
3450
H. Cheng et al.
Fig. 14. Plots of d234U vs. [231Pa/235U] (a) and d234U vs. [230Th/238U] (b) for the last interglacial samples. The symbols
are as same as in Figs. 10 and 13. Best fit lines are y 5 428.9x 2283.9 and y 5 396.0x 2192.3 in diagrams a and b,
respectively. The best-fit lines intersect the concordia curves (d234U 5 150) at an ages of 116 ka (diagram a) and 121 ka
(diagram b). We consider both of these age estimates to be less reliable than the estimate from Fig. 13b because (1) data
points in both plots lie off of a best-fit line, and (2) the error in age due to uncertainty in initial d234U is significant in both
plots.
line translate into larger errors in time for the [231Pa/235U] vs.
[230Th/238U] plot. For all of these reasons, the intersection of
concordia with the best-fit line in the [231Pa/235U] vs. [230Th/
234
U] provides the best estimate of primary age for this particular example.
4. CONCLUSIONS
With precise 230Th and 231Pa TIMS data now available,
testing for U-series concordancy should become common.
Samples with concordant 230Th and 231Pa ages will be likely
candidates for closed-system histories. For nonmarine samples,
such as speleothems, teeth, and bones, 231Pa TIMS methods
might provide the only possible high-precision check for 230Th
dating. For marine samples, the initial d234U value can also be
compared to the modern marine value (about 150). If samples
give concordant 230Th and 231Pa ages but values of initial
d234U different from the modern marine value, this could
indicate a different initial d234U value for past seawater, or
suggest a type of diagenesis which resulted in the 234U change
without disturbing the 230Th and 231Pa ages, for example, early
234
U input or 231Pa, 230Th, and 234U input in just the right
proportions (Figs. 9 and 10).
When a single sample has discordant 230Th and 231Pa ages,
it can generally be considered to have experienced open-system
behavior (assuming no significant initial 230Th and 231Pa). To
provide some constraints on its true age, a [231Pa/235U] vs.
[230Th/234U] diagram might be used in an effort to reduce the
effects of disturbed d234U values. If the isotopic composition
lies above the concordia curve in a [231Pa/235U] vs. [230Th/
234
U] diagram, it is likely that the dominant diagenetic process
was U loss or gain of both 230Th and 231Pa. If the isotopic
composition lies below the concordia curve, then U gain or loss
of both 230Th and 231Pa is probable (Figs. 12Fig. 10). In the
former case, its true age will be younger than its apparent 230Th
and 231Pa ages. In the later case, its true age will be older.
231
Pa/230Th ages should be closest to the true age in such cases.
The use of U-Th-Pa systematics can also provide checks on
the LU or EU models, important to many tooth and bone dating
studies relating to archaeology. When the true age is greater
than 20 ka, in the LU model, the measured data should lie
below the concordia curve. Thus, this approach can be used to
check the veracity of model 231Pa and 230Th ages based on
different assumed modes of U uptake. Furthermore, such data
may be useful in establishing constraints on U uptake histories
for specific samples that have aged in particular diagenetic
environments. Using these sample specific constraints on U
uptake, it may then be possible to constrain the true age of such
an open-system material.
As noted, the use of concordia diagrams has been a unique
strength of U-Pb dating. A single discordant U-Pb date is not
very meaningful, instead, the upper intercept point of a series of
samples becomes important for an age determination. A similar
approach can be used in the U-Th-Pa dating of discordant
samples. For example, a suite of samples of the same age (for
example, corals from the same terrace and location) can be used
to provide a series of points for a concordia diagram. If these
samples experienced different degrees of one of the diagenetic
processes described by the above models, the data will fall
U-Th-Pa dating
along a straight line or a curve that is very close to a straight
line near the upper intercept, in a concordia diagram. The upper
intercept will be the true crystallization age.
Acknowledgments—We thank J. A. Hoff for discussions on this research. This work was supported by NSF grants OCE-9402693, OCE9500647, EAR-9512334, EAR-9406183, and EAR-9702137 to RLE,
the Donors of the Petroleum Research Fund, administered by the
American Chemical Society, and a grant to MTM from the Geosciences
Research Program, Office of Basic Energy Sciences, U.S. Department
of Energy.
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