_{1}

^{*}

The slope year
t_{slope}
for the U-Pb dating method is given as
,
where λ_{238} and λ_{235} are the decay constants for ^{238}U and ^{235}U, respectively, and k is the slope of the tangent line at a point on either the Concordia or Discordia line. These two lines are determined by the initial ^{206(7)}Pb_{i} concentrations in minerals. If , the line is the Concordia. However, if (∧ is the logical operator “and”, also known as the logical conjunction), or , the line is Discordia. The Concordia line is of the form (where p stands for the present), while the Discordia line has the form (where k and b are the slope and intercept of the straight line, respectively).

In nature, uranium has three radioactive isotopes: ^{238}U(99.2743%), ^{235}U(0.7200%) and ^{234}U(0.0057%) [

and

where Q is the heat, β denotes the beta decay and He stands for the element Helium. The decay constants λ for ^{238}U and ^{235}U are ^{−1} and ^{−1}, respectively [

These nuclear reactions occur in host minerals, such as zircon (ZrSiO_{4}), and are the basis of the U-Pb dating method in geology [

and

where the subscripts i and p represent the initial measurement time and the present, respectively, and t is the age of the mineral [

The coordinates n(^{206}Pb_{p})/n(^{238}U_{p}) (n, the number of isotopes in the bracket) as the ordinate and n(^{207}Pb_{p})/ n(^{235}U_{p}) ratios as the abscissa form the Pb/U ratio diagram (^{206}Pb_{p})/n(^{238}U_{p}):

Similarly for n(^{207}Pb_{p})/n(^{235}U_{p}), we have

from Equation (2).

To interpret the Discordia line, conventional theories have proposed: 1) this line was caused by Pb loss or U gain after formation of the host mineral [

However, previous theories are not tenable when used in the following cases:

1) the lower intercept point is negative or

2) no upper intercept point exists.

For instance, in Zheng et al. (2012) (

Herein, the slope years t_{slope}s for the U-Pb dating method for the Concordia and Discordia lines are presented, and a method for estimating values for t_{slope} from the experimental data is proposed. In addition, four examples are presented to illustrate the application of the proposed method.

In this study, the basic assumptions for the U-Pb dating method included the following:

a) The decay constants λ_{238} and λ_{235} are precisely determined. For instance, the decay constants in Jaffey et al. (1971) are of good quality and widely accepted. The number of citations of this paper is greater than 1200 (data from Web of Science);

b) Host minerals are not influenced by chemical reactions after formation. The minerals included apatite [

c) Present ^{206(7)}Pb_{p} and ^{235(8)}U_{p} isotope concentrations in host minerals can be precisely measured using mass spectrometry (MS). Such MS instruments include sensitive high mass-resolution ion microprobe (SHRIMP) [

In mathematics, the variance on the ordinate is a function of the variance on the abscissa [^{206}Pb_{p})/n(^{238}U_{p}) is a function of n(^{207}Pb_{p})/n(^{235}U_{p}) in the Pb/U diagram (

The theoretical expressions for this function under different conditions are given in Section 2.4.

Next, the slope k of the tangent line at point A on the general curve of Equation (5) was determined. The partial derivative of

Similarly, we have

from Equation (4). Equation (6) divided by Equation (7) gives

In this equation, the second part is the definition of the slope of the tangent line [

This equation indicates that if t is determined, the value of k is a constant (

If k is determined (see Section 2.6), the slope year is given by rewriting Equation (9):

If the values for t_{slope}, ^{206(7)}Pb_{p} and ^{235(8)}U_{p} are known, the initial ^{206(7)}Pb_{i} concentrations in minerals can be determined using the following:

and

which are derived from Equations (1) and (2). Clearly, the concentrations are greater than or equal to zero:

The initial ^{206(7)}Pb_{i} isotope concentrations determine the mathematical expressions for the general graph in

and an additional three samples (4, 5 and 6,

t(Ma) | 0 | 100 | 1000 | 2000 | 3000 | 4000 | 5000 |
---|---|---|---|---|---|---|---|

k^{a} | 0.15751 | 0.14497 | 0.06870 | 0.02997 | 0.01307 | 0.00570 | 0.00249 |

a, calculated from Equation (9)

The mathematical expressions are given by solving the first-order differential Equation (9) using Equations (3) and (4):

The solution to this equation is different for each set of samples.

a) For samples 1, 2 and 3, rewriting Equation (15) using Equation (13) gives

The general solution of Equation (16) is

Since the concentrations of

or

which is the expression for the Concordia line.

b) For samples 4, 5 and 6, because of the existence of the variances in

This difficulty can be overcome in the following manner. Consider a geological body (containing samples 4, 5 and 6) with continuous ^{206}Pb_{i}, ^{207}Pb_{i},

Since k is a constant when t is given (

where k and b are the slope and intercept of the line, respectively. This equation shows that the general curve in

Equation (21) is consistent with the initial condition (Equation (14)). If k = 0.15751 (at t = 0) is applied:

This equation indicates that 1) in the geological system, ^{206}Pb_{i}/^{238}U_{i} monotonically increases with increasing ^{207}Pb_{i}/^{235}U_{i} from samples 4 to 5 to 6 (

The

a) samples 1, 2 and 3 (

b) samples 4, 5 and 6 (

For n

a) If the n data points plot on the Concordia line (

where

b) If the n data points plot on the Discordia line (

where

and_{Discordia} in Appendix A.

For a function

where

According to Equation (27), the standard error for t_{slope} (Equation (10)) is

or

where ^{−1} and ^{−1} [

a) For concordant data, the standard error of the ith slope (Equation (23)) is

and the standard error of the mean slope (Equation (24)) is

b) For discordant data, the standard error of k in Equation (26) is

See proofs of this equation in Appendix A.

According to Equation (27), the standard error for ^{206(207)}Pb_{i} (Equations (11) and (12)) is

where m and n stand for 206(7) and 235(8) respectively, ^{−1} and ^{−1} [

To demonstrate the validity of our work, four examples are illustrated (

The first example comes from Qinghu granite in the Nanling Range, South China [_{Concordia} = 0.13792 ± 0.00025 and t_{slope} = 160 ± 2 Ma (

The k and t_{slope} values for the three discordant examples described in the introduction were also calculated using Equations (26) and (10), respectively. For the Zimbabwe uranium deposit (_{Discordia} = 0.03950 ± 0.00178 and slope year was t_{slope} = 1668 ± 55 Ma. For amphibolites in the Yingxian lamproite (YX1, _{Discordia} = 0.06779 ± 0.00564 and slope year was t_{slope} = 1016 ± 100 Ma. For Hebi amphibolites (HBxa, _{Discordia} = 0.010734 ± 0.00196 and slope year was t_{slope} = 3237 ± 220 Ma.

A method for determining the slope year for the U-Pb dating method and initial ^{206(7)}Pb concentrations in samples was described. It was also found that if no ^{206(7)}Pb isotopes are initially present in minerals, the Pb/U ratios plot on the Concordia line. On the other hand, if ^{206(7)}Pb isotopes are initially present in minerals, the Pb/U ratios plot on the Discordia line. Therefore, the Discordia line is not the result of Pb loss or U gain. Furthermore, methods for determining the slope year using experimental data were also proposed and applied to data on four samples previously described in the literature. These results demonstrate that our approach is useful for geological research.

Type | Experiments | Present Results | |||||||
---|---|---|---|---|---|---|---|---|---|

Locations | Samples | ^{206}Pb/^{238}U | 1σ | ^{207}Pb/^{235}U | 1σ | Item | Value | 1σ | |

Concordia | 07QH-1 | 1 | 0.0250 | 0.0003 | 0.171 | 0.003 | k | 0.13792 | 0.00025 |

2 | 0.0253 | 0.0003 | 0.172 | 0.002 | t_{slope } | 160 | 2 Ma | ||

3 | 0.0252 | 0.0003 | 0.172 | 0.003 | |||||

4 | 0.0250 | 0.0003 | 0.170 | 0.003 | |||||

5 | 0.0252 | 0.0003 | 0.172 | 0.002 | |||||

6 | 0.0249 | 0.0003 | 0.171 | 0.003 | |||||

7 | 0.0251 | 0.0003 | 0.173 | 0.002 | |||||

8 | 0.0251 | 0.0003 | 0.168 | 0.003 | |||||

9 | 0.0251 | 0.0003 | 0.176 | 0.003 | |||||

10 | 0.0251 | 0.0003 | 0.170 | 0.003 | |||||

11 | 0.0251 | 0.0003 | 0.172 | 0.002 | |||||

12 | 0.0248 | 0.0003 | 0.169 | 0.003 | |||||

13 | 0.0251 | 0.0003 | 0.172 | 0.003 | |||||

14 | 0.0250 | 0.0003 | 0.170 | 0.002 | |||||

15 | 0.0250 | 0.0003 | 0.169 | 0.002 | |||||

16 | 0.0249 | 0.0003 | 0.166 | 0.002 | |||||

17 | 0.0250 | 0.0003 | 0.171 | 0.002 | |||||

18 | 0.0249 | 0.0003 | 0.170 | 0.002 | |||||

19 | 0.0249 | 0.0003 | 0.168 | 0.002 | |||||

20 | 0.0252 | 0.0003 | 0.174 | 0.002 | |||||

Mean | 0.0250 | 0.0003 | 0.171 | 0.0025 | |||||

Discordia | Zimbabwe | Monazite(Manitoba) | 0.634 | 0.000 | 14.75 | 0.00 | k | 0.03950 | 0.00178 |

Monazite(Ebonite) | 0.507 | 0.000 | 12.45 | 0.00 | t_{slope} | 1667 | 55 Ma | ||

Monazite(Jack Tin) | 0.420 | 0.000 | 10.10 | 0.00 | |||||

Monazite(Irumi) | 0.383 | 0.000 | 9.02 | 0.00 | |||||

Uraainite(Manitoba) | 0.270 | 0.000 | 5.85 | 0.00 | |||||

Monazite(Antsirabe) | 0.241 | 0.000 | 5.16 | 0.00 | |||||

Discordia | YX1 | 1 | 0.33464 | 0.00363 | 5.22129 | 0.06472 | k | 0.06779 | 0.00564 |

2 | 0.34491 | 0.00368 | 5.52554 | 0.06520 | t_{slope} | 1016 | 100 Ma | ||

3 | 0.33249 | 0.00385 | 5.12718 | 0.07519 | |||||

4 | 0.33347 | 0.00352 | 5.19461 | 0.05960 | |||||

5 | 0.21231 | 0.00226 | 3.66393 | 0.04298 | |||||

6 | 0.33912 | 0.00358 | 5.34786 | 0.06130 |

7 | 0.33246 | 0.00353 | 5.22593 | 0.06103 | ||||||
---|---|---|---|---|---|---|---|---|---|---|

8 | 0.24655 | 0.00268 | 3.94621 | 0.04940 | ||||||

9 | 0.30931 | 0.00328 | 5.33072 | 0.06161 | ||||||

10 | 0.26968 | 0.00309 | 4.22308 | 0.05705 | ||||||

11 | 0.34094 | 0.00374 | 5.29417 | 0.06276 | ||||||

Discordia | Hbxa | 1 | 0.31857 | 0.00389 | 6.52978 | 0.08636 | k | 0.010734 | 0.001956 | |

2c | 0.37868 | 0.00542 | 11.83149 | 0.19556 | t_{slope} | 3237 | 220 Ma | |||

2r | 0.35917 | 0.00452 | 9.43864 | 0.13599 | ||||||

3c | 0.32201 | 0.00375 | 6.60397 | 0.08699 | ||||||

3r | 0.32726 | 0.00388 | 6.39918 | 0.08482 | ||||||

4 | 0.35923 | 0.00457 | 10.94269 | 0.15946 | ||||||

5 | 0.33858 | 0.00402 | 8.05032 | 0.10509 | ||||||

6 | 0.32256 | 0.00368 | 6.04709 | 0.07394 | ||||||

7 | 0.32507 | 0.00396 | 6.51079 | 0.08888 | ||||||

8 | 0.29783 | 0.00355 | 6.74590 | 0.09045 | ||||||

9 | 0.32970 | 0.00477 | 7.19338 | 0.12913 | ||||||

10 | 0.34275 | 0.00486 | 8.54210 | 0.14216 | ||||||

11 | 0.31630 | 0.00412 | 7.75610 | 0.12353 | ||||||

12 | 0.30213 | 0.00442 | 7.02483 | 0.12968 | ||||||

13 | 0.31948 | 0.00461 | 6.22754 | 0.13341 |

This work was supported by the National Natural Science Foundation of China (Grant Nos. 41303047, 90914010 and 41020134003).

The least squares method is described in textbooks on probability statistics [_{1}, y_{1},) … (x_{n}_{,}y_{n}), there is a line:

that best fits the data. The quality of this line is determined by

When

To find the minimum value for

giving

where

The variance of a new predicted

where σ is the standard error of

if n is very small. Because k follows a Gaussian distribution, its variance is

The square root of this equation is the 1σ error of k.